Cracking The Ap Calculus ab Exam 2018

able to do that step on your own. Example 3: (^) ∫ 3 sin(3x − 1) dx = Let u = 3x − 1. Then d ...

Example 5: (^) ∫ x cos(3x^2 + 1) dx = Let u = 3x^2 + 1. Then du = 6x dx. We need to substitute fo ...

This is a good technique to master, so practice on the following solved problems. Do each problem, cov ...

(4x^2 − 7)^11 + C PROBLEM 4. Evaluate ∫ tan sec^2 dx. Answer: Let u = tan and du = sec^2 dx. ...

7. 8.∫ sin(sin x)cos x dx ...

Chapter 14 Definite Integrals ...

AREA UNDER A CURVE It’s time to learn one of the most important uses of the integral. We’ve alread ...

Notice that the width of each rectangle is 1. The height of the left rectangle is found by plugging ...

Now find the height of each rectangle the same way as before. Notice that the values we use are the left end ...

To find the area of the rectangles, we would still use the width of , but the heights would change. The heig ...

we use equal to n. So the width of each rectangle is . The height of the first inscribed rectangle is ...

dividing the region into four rectangles. The region would look like the following: To find the area o ...

(Note: The fractional subscript means to evaluate the function at the number halfway between ...

The width of each rectangle is . The heights of the rectangles are 23 , , , and Therefore, the area is ...

Example 3: Repeat Example 1 using four midpoint rectangles. Now draw four rectangles that look like ...

f(x). These are quite simple to evaluate. All you do is use the right-hand or left-hand sum ...

replace the values of with the appropriate widths of each rectangle. The width of the first r ...

Because = 16.25, you can see how close we were with our three earlier approximations. Example 6: Find (x^ ...

We get a picture that looks like the following: As you should recall from geometry, the formula for the area ...

Notice how each trapezoid shares a base with the trapezoid next to it, except for the end ones. This ena ...