Exam 5: Integration

List the evaluation points corresponding to the midpoint of each subinterval, sketch the function and corresponding approximating rectangles and evaluate the corresponding Riemann sum. Round the sum to two decimal places.

Use Part I of the Fundamental Theorem of Calculus to compute the integral exactly.

Write out all terms and compute the sum.

Use the Midpoint Rule to estimate the value of the integral (obtain two digits of accuracy).

Graph the function .

Determine the number of steps to guarantee an accuracy of when approximating the integral using (a) the Trapezoidal Rule, (b) Midpoint Rule, and (c) Simpson's Rule.

Use Riemann sums and a limit to compute the exact area under the curve. on

Find the area of the region bounded by , , the x-axis, and the y-axis.

Find the derivative

Evaluate the integral.

Use the given information about and its derivatives to determine whether (a) the Midpoint Rule would be exact, underestimate or overestimate the integeral (or if there's not enough information to tell). Repeat for (b) the Trapezoidal Rule and (c) Simpson's Rule.

Use Riemann sums and a limit to compute the exact area under the curve. on

Use Part I of the Fundamental Theorem of Calculus to compute the integral exactly.

Use mathematical induction to prove that for all integers .

Find the function satisfying the given conditions.

Evaluate the integral.

Use the properties of logarithms to rewrite the expression as a single term.

Evaluate the integral.

Make the indicated substitution for an unspecified function . for

Evaluate the derivative using properties of logarithms where needed.