5 Steps to a 5 AP Calculus BC 2019

292 STEP 4. Review the Knowledge You Need to Score High

Example 2

Find the length of the curver(t)=

〈

2 sint,5t

〉

fromt=0tot=π.

Step 1. r′(t)=

〈

2 cost,5

〉

Step 2.

∥∥

r′(t)

∥∥

=

√

4 cos^2 t+ 25

Step 3. With the aid of a graphing calculator, the arc lengths=

∫π

0

√

4 cos^2 t+ 25 dtcan

be found to be approximately equal to 16.319 units.

12.6 Rapid Review

1. Iff(x)=

∫x

0

g(t)dtand the graph ofgis shown in Figure 12.6-1. Find f(3).

y

g(t)

0 321 t

–1

1

Figure 12.6-1

Answer: f( 3 )=

∫ 3

0

g(t)dt=

∫ 1

0

g(t)dt+

∫ 3

1

g(t)dt

= 0. 5 − 1. 5 =− 1

2. The functionfis continous on [1, 5] and f >0 and selected values of fare given
   below.

x 1 2 3 4 5

f(x) 2 4 6 8 10

Using 2 midpoint rectangles, approximate the area under the curve offforx=1to

x=5.

Answer:Midpoints arex=2 andx=4 and the width of each rectangle

=

5 − 1

2

=2.

Area≈ Area of Rect 1 + Area of Rect 2 ≈4(2)+8(2)≈24.

3. Set up an integral to find the area of the regions bounded by the graphs ofy=x^3 and
   y=x. Do not evaluate the integral.