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mathematics
precalculus

Questions and Answers of Precalculus

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. = y°, x = 1 – y²;
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. ху — 1, у %3 0, х
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. about x — 2 у 3 х*,
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. sin x, y = cos x, 0
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. = 1 sec x, y = 3; about
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.about y = -3 y = x', y =
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.about y = 1 у 3D х?, х
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.about the y-axis x = 2 -
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.about the y-axis y = 6
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.about the y-axis — х*,
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.about the y-axis 2х 3D
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.about the y-axis х 3
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.about the y-axis y = e*,
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.about the y-axis У3 Vx -
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.about the y-axis у%3D
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.about the y-axis у 3 х
The rates at which rain fell, in inches per hour, in two different locations t hours after the start of a storm are given by andCompute the area between the graphs for 0 < t < 2 and
In Example 5, we modeled a measles pathogenesis curve by a function f. A patient infected with the measles virus who has some immunity to the virus has a pathogenesis curve that can be modeled by,
If the birth rate of a population is b(t) = 2200e0.024t people per year and the death rate is d(t) = 1460e0.018t people per year, find the area between these curves for 0 < t < 10. What does
Graph the region between the curves and use your calculator to compute the area correct to five decimal places. y = cos x, y = x + 2 sin4x
Graph the region between the curves and use your calculator to compute the area correct to five decimal places. y = tan2x, y = √x
Graph the region between the curves and use your calculator to compute the area correct to five decimal places. y = e1-x2, y = x4
Graph the region between the curves and use your calculator to compute the area correct to five decimal places. y = 2/1 + x4, y = x2
Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. y = 1.3x, y = 2√x
Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves. y = 3x2 - 2x, y = x3 - 3x
Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.y = x/(x2 + 1)2, y = x5, x >
Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the area of the region bounded by the curves.y = x sin (x2), y = x4, x > 0
Evaluate the integral and interpret it as the area of a region. Sketch the region. 3* – 2* | dx L 1
Use calculus to find the area of the triangle with the given vertices. (2, 0), (0, 2), (-1, 1)
Sketch the region enclosed by the given curves and find its area.y = ln x/x, y = (ln x)2/x
Sketch the region enclosed by the given curves and find its area. х y : 1 + x³
Sketch the region enclosed by the given curves and find its area. х х y = 19 — х2 y = V1 + x? х>0
The graphs of two functions are shown with the areas of the regions between the curves indicated.(a) What is the total area between the curves for 0 < x < 5?(b) What is the value of J
Sketch the region enclosed by the given curves and find its area. у — x*, у — 2х*, х+ у— 3, х>0
Sketch the region enclosed by the given curves and find its area. х > 0 у — х, у 3 1/х, у—+х,
Sketch the region enclosed by the given curves and find its area.y = sinh x, y = e-x, x = 0, x = 2
Sketch the region enclosed by the given curves and find its area.y = cos x, y = 1 - cos x, 0 < x < π
Sketch the region enclosed by the given curves and find its area. y=x², 0
Sketch the region enclosed by the given curves and find its area. ytan x, y = 2 sin x, /3 x /3
Sketch the region enclosed by the given curves and find its area.x = y4, y = √2 - x, y = 0
Sketch the region enclosed by the given curves and find its area.y = cos πx, y = 4x2 - 1
Sketch the region enclosed by the given curves and find its area. y = √x - 1, x - y = 1
Sketch the region enclosed by the given curves and find its area.x = 2y2, x = 4 + y2
Sketch the region enclosed by the given curves and find its area.y = cos x, y = 2 - cos x, 0 < x < 2π
Sketch the region enclosed by the given curves and find its area. y = sec?x, y= 8 cos x,
Sketch the region enclosed by the given curves and find its area.y = x2, y = 4x - x2
Sketch the region enclosed by the given curves and find its area.y = 12 - x2, y = x2 - 6
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle and label its height and width. Then find the area of the
Find the area of the shaded region. УА 4y (-3, 3) х х%3D2у — у?
Find the area of the shaded region. Ул x= y? – 2 y=1 (x= e y=-1
Find the area of the shaded region. ул П.o (1, e) y= e* у-хе У3х х
Find the area of the shaded region. yA y= {k (1, 1) х—8 х y= 1/x
Evaluate 1 Vn n + 2 In Vn + n lim Vn + 1
Given the point (a, b) in the first quadrant, find the downward-opening parabola that passes through the point (a, b) and the origin such that the area under the parabola is a minimum.
A circular disk of radius r is used in an evaporator and is rotated in a vertical plane. If it is to be partially submerged in the liquid so as to maximize the exposed wetted area of the disk, show
Evaluate | (1 – tan 21)'/' dt. lim х—0 х Jо
Suppose f is continuous, f (0) = 0, f (1) − 1, f'(x) > 0, andFind the value of the integral
Evaluate :(e)-(4) - (A)- -() 9. 9. 3 п lim
If f is continuous on [0, 1], prove that f(x) dx = f(1 – x) dx
Find *2+h lim h→0 h J2 V1 + t3 dt
If f' is continuous on [a, b], show that f(x)f'(x) dx = [f(b)]² – [f(a)I²
Suppose h is a function such that
If f is a continuous function such thatfor all x, find an explicit formula for f (x). х F() dt = (x – 1)e²* + f(t) dt Se , 2x
Suppose that the temperature in a long, thin rod placed along the x-axis is initially C/(2a) if |x | < a and 0 if |x | > a. It can be shown that if the heat diffusivity of the rod is k, then
The Fresnel function was introduced in Section 5.3. Fresnel also used the functionin his theory of the diffraction of light waves.(a) On what intervals is C increasing?(b) On what
Let r(t) be the rate at which the world's oil is consumed, where t is measured in years starting at t = 0 on January 1, 2000, and r(t) is measured in barrels per year. What does ∫80 r(t) dt
Use the properties of integrals to verify the inequality. x sin 'x dx
Use the properties of integrals to verify the inequality. e* cos x dx
Use the properties of integrals to verify the inequality. /2 dx < *T/2 sin x T/4 х VI
Use the properties of integrals to verify the inequality. x² cos x dx< 3 VI
Use Property 8 of integrals to estimate the value of the integral. *5 - dx Јз х+ 1
Find the derivative of the function. sin(t*) dt *3x+1 y = 2x
Find the derivative of the function. e' dt х y
Find the derivative of the function.
Find the derivative of the function. g(x) | cos(r*) dt ||
Find the derivative of the function. F(x) = [ -Vt + sin t dt
Find the derivative of the function. F(x) = |* dt .3 Jo 1 + t³
Evaluate the integral, if it exists. Vx – 1| dx
Evaluate the integral, if it exists. x² – 4| dx '3
Evaluate the integral, if it exists. *T/4 (1 + tan t)³ sec?t dt
Evaluate the integral, if it exists. sec 0 tan 0 do 1 + sec 0
Evaluate the integral, if it exists. sinh(1 + 4x) dx
Evaluate the integral, if it exists. x3 -dx- 1 + x4
Evaluate the integral, if it exists. х dx 1 — х4
Evaluate the integral, if it exists. tan x In(cos xr) dx
Evaluate the integral, if it exists. sin(ln x) dx
Evaluate the integral, if it exists. evi dx х
Evaluate the integral, if it exists. sin x cos(cos x) dx
Evaluate the integral, if it exists. sin Tt cos TTt dt
Evaluate the integral, if it exists. csc?x csc-x dx 1 + cot x
Evaluate the integral, if it exists. х+2 dx 4x х
Evaluate the integral, if it exists. 2 - х dx х
Evaluate the integral, if it exists. e* dx lo 1 + e2x

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