ST 552

Calculus on a Computer

Assignment 9 – Integrals: Areas as Sums

1. Approximate the area under the curve between x=2 and x=4 using 10 rectangles and

a. The left endpoints

b. The right endpoints

c. The midpoints

2. Repeat exercise 1 with the upper half of the circle . Compare to the exact answer

3. Prove that .

4. The trapezoid formula approximates an area by using a straight line on each subinterval, as shown below. This gives tall skinny trapezoids. Write out the general formula for the trapezoid formula to approximate the area under f(x) from x=a to x=b. Now, for n=5, expand out the sum. What do you see?

5. Use the trapezoid formula with n=8 to approximate .