In this question, you will estimate the area under the curve

y= 4x³

fromx= 3 tox= 5 using three different Riemann sums.

You will subdivide the interval [3,5] into four sub-intervals of equal width.

1. Using our standard notation for Riemann sums, enter the values ofa,b,n, andx. a=b=n=x=
2. Complete the following table listing the four sub-intervals:

   First subinterval	: [,]
   Second subinterval	: [,]
   Third subinterval	: [,]
   Fourth subinterval	: [,]

3. In the next two parts of the question, you will calculate the approximate area under the curve using theleftend-points of the sub-intervals. Complete the following table

   x1	:	f(x1)	:
   x2	:	f(x2)	:
   x3	:	f(x3)	:
   x4	:	f(x4)	:

4. Now calculate the approximate area under the curve using the formula Approximate area under curve :
5. In the next two parts of the question, you will calculate the approximate area under the curve using therightend-points of the sub-intervals. Complete the following table (for the second column, you should only have to do one calculation from scratch, since you have already calculated three of the numbers earlier in the question).

   x1	:	f(x1)	:
   x2	:	f(x2)	:
   x3	:	f(x3)	:
   x4	:	f(x4)	:

6. Now calculate the approximate area under the curve using the formula Approximate area under curve :
7. In the next two parts of the question, you will calculate the approximate area under the curve using themid-pointsof the sub-intervals. Complete the following table

   x1	:	f(x1)	:
   x2	:	f(x2)	:
   x3	:	f(x3)	:
   x4	:	f(x4)	:

8. Now calculate the approximate area under the curve using the formula Approximate area under curve :
9. You can calculate the exact area under the curve using the Fundamental Theorem of Calculus: whereF(x) is an antiderivative of the functionf(x) = 4x³. Find an antiderivative of the functionf(x). Enter your constant of integration asc.Antiderivative : Now find the exact value of the area under the curve.Area =Notice that your value ofcdoes not matter when you calculate the area under the curve.

You should notice that the left end-point calculation under-estimates the area and that the right end-point calculation over-estimates the area. This is because the function is increasing.