5 Steps to a 5 AP Calculus BC 2019

Areas, Volumes, and Arc Lengths 279

Example 3

The base of a solid is the region enclosed by a triangle whose vertices are (0, 0),

(4, 0), and (0, 2). The cross sections are semicircles perpendicular to thex-axis. Using a

calculator, find the volume of the solid. (See Figure 12.4-4.)

2

0

4

y

x

Figure 12.4-4

Step 1. Find the area of a cross section.

Equation of the line passing through (0, 2) and (4, 0):

y=mx+b;m=

0 − 2

4 − 0

=−

1

2

;b= 2

y=−

1

2

x+ 2.

Area of semicircle =

1

2

πr^2 ;r=

1

2

y=

1

2

(

−

1

2

x+ 2

)

=−

1

4

x+ 1.

A(x)=

1

2

π

(

y

2

) 2

=

π

2

(

−

1

4

x+ 1

) 2

.

Step 2. Set up an integral.

V=

∫ 4

0

A(x)dx=

∫ 4

0

π

2

(

−

1

4

x+ 1

) 2

dx

Step 3. Evaluate the integral.

Enter

∫ ((

π

2

)

∗(−. 25 x+ 1 )∧2, x,0,4

)

and obtain 2.0944.

Thus the volume of the solid is 2.094.

TIP • Remember: iff′<0, then f is decreasing, and if f′′<0 then the graph of f is
concave downward.