Because = 16.25, you can see how close we were with our three earlier approximations.
Example 6: Find (x^2 + 2) dx.
The Fundamental Theorem of Calculus yields the following:
(x^2 + 2) dx = If we evaluate this, we get the following:
This is the first function for which we found the approximate area by using inscribed rectangles. Our final
estimate, where we averaged the inscribed and circumscribed rectangles, was , and as you can see, that
was very close (off by ). When we used the midpoints, we were off by .
We’re going to do only a few approximations using rectangles, because it’s not a big part of the AP Exam.
On the other hand, definite integrals are a huge part of the rest of this book.
Example 7: (x^2 − x) dx = =
Example 8: sin x dx = (−cos x) − (−cos 0) = 1
Example 9: sec^2 x dx = tan x = tan − tan 0 = 1
The Trapezoid Rule
There’s another approximation method that’s even better than the rectangle method. Essentially, all you do
is divide the region into trapezoids instead of rectangles. Let’s use the problem that we did at the
beginning of the chapter.