For the integral, (a) find the indicated approximation, and(b) determine if your approximation is an under-estimate, an over-estimate, or exact.

- TRAP(4) for

Answer

- This time we get sub-intervals of length 5. We can see our answer to (c) from the picture.

- Since a straight line has no concavity, the trapezoid rule gives the exact value of the integral. We can verify this by finding the areas of the triangles above and below the
*x*-axis and calculating the value of the integral exactly. The line crosses the *x*-axis when *x* = 6. The big triangle has base 16 and "height" *f* (-10) = -8, so its weighted area is -64.

The little triangle has base 4 and height *f* (10) = 2, so its weighted area is 4. Combining these, we get

which confirms that the trapezoid rule gave us the exact value of the integral.