a. Calculate the volume of the solid of revolution created by rotating the curve y = 2 + 3exp (-3 x) about the x-axis, for x between 2 and 3. Volume : b. The equation of a circle of radius r, centered at the origin (0,0), is given by 12 = x2+ y- o Rearrange this equation to find a formula for y in terms of x and r. (Take the positive root.) Equation: y = . What solid of revolution is swept out if this curve is rotated around the x axis, and x is allowed to vary between -r and r? (You do not need to enter this answer into WebAssign.) o Suppose we wanted to set up the following integral so that V gives the volume of a sphere of radius / v = f (x) dx What would a, b and f(x) be? a = b = f ( x ) = (WebAssign note: remember that you enter IT as pi ) o Carry out the integration, and calculate the value of V in terms of r. V =In this question, you will use a substitution to carry out the following integration: (23 + 1) 3x2 dx If the answer requires a constant of integration, enter it as c. a. The integral involves the composite function (2" + 1)" . What u-substitution will simplify this term? U = g(x) = du b. Find the derivative dax : du dx = c. Transform the original integral into one involving u by using the substitution rule: o Replace all occurrences of 9 ( ) in the integral by U. du Replace 9 (x) dx by du ( equivalently: replace d. by du ). du (you do not need to enter du in your answer) d. Carry out the integration, and find the most general antiderivative (in terms of u). antiderivative : e. Finally, rewrite your answer in terms of x by replacing u by g(x). (23 + 1) 3x3 dx * symbolic formatting help Submit Answer 2. [-/12 Points] DETAILS MY NOTES Integrate each of the following functions using substitution, finding the most general antiderivative. Also enter u, the function of x that you substitute. If your answer requires a constant of integration, enter it as c. 3x Vx3 + 1 da U = 1x3 +4 dx b. Vx4 + 4x + 4 U = (In x)' dx U = d. 25 (2x6 + 2)" dx U= dx e. J V5x3 + 5 U = 3x7 +5 dx U = symbolic formatting helpIn this question, you will estimate the value of the integral xe ? dx using three different approximations. a. Subdivide the interval [1,7] into three sub-intervals of equal width and complete the following: Ax = f (ao ) = f(a 1 ) = a, = f ( a2 ) = a3 = f (a3 ) = X1 = f ( x 1 ) = X2 = f ( x 2 ) = X3 = f ( x 3 ) = b. Calculate the approximate value of the integral using the trapezoidal rule. Area ~ c. Calculate the approximate value of the integral using the midpoint rule. Area ~ d. Calculate the approximate value of the integral using Simpson's rule. Area ~ e. It is possible to show that an antiderivative of x e-*/2 is -2 (x + 2) ez Using this antiderivative, calculate the exact value of the integral. Integral = Submit Answer [-/4 Points] DETAILS MY NOTES In this question, you will investigate whether the improper integral int_1 ^infinity 1/x^2 text( )dx converges or diverges. If it converges, you will find its value. a. Calculate the value of the integral int_1^(b) 1/x^2 text( )dx where b is a finite number whose value is greater than one. Value = b. Does the value of the integral approach a limit as b tends to infinity? If so, enter this limiting value: int_1^infinity 1/x^2 text( )dx symbolic formatting help