5 Steps to a 5 AP Calculus BC 2019

Areas, Volumes, and Arc Lengths 307

(c)a(t)=v′(t)

Enterd(x∗cos(x∧ 2 +1), x)|x=2 and

obtain 7.95506.

Thus, the velocity of the particle is

increasing att=2, sincea(2)>0.

30. (See Figure 12.10-6)

[−π,π] by [−1,2]

Figure 12.10-6

(a)Point of Intersection: Use the

[Intersection] function of the calculator

and obtain (0.517757, 0.868931).

Area =

∫ 0. 51775

0

(cosx−xex)dx

Enter

∫

(cos(x)−x∗e∧x, x,

0, 0. 51775 )and obtain 0.304261.

The area of the region is approximately

0.304.

(b) Step 1. Using the Washer Method:

Outer radius=cosx;

Inner radius=xex.

V=π

∫ 0. 51775

0

[

(cosx)^2 −(xex)^2

]

dx

Step 2. Enter

∫ (

π((cos(x)∧ 2 )−

(x∗e∧(x))∧ 2

)

, x,0. 51775

)

and obtain 1.16678.

The volume of the solid is

approximately 1.167.

31. Convert to a parametric representation
    withx=rcosθ=5 cosθcos 2θand
    y=rsinθ=5 cos 2θsinθ. Differentiate
    with respect toθ.
    dx
    dθ
    =−5 cos 2θsinθ−10 sin 2θsinθand

dy

dθ

=5 cos 2θcosθ−10 sin 2θsinθ.

Divide to find

dy

dx

=

5 cos 2θsinθ−10 sin 2θsinθ

−5 cos 2θsinθ−10 sin 2θsinθ

=

−cos 2θcosθ+2 sin 2θsinθ

cos 2θsinθ+2 sin 2θsinθ

. Evaluated

atθ=

3 π

2

,

dy

dx

=0.

The slope of the tangent line is zero,

including a horizontal tangent.

32.

∫

2

x^2 − 4 x

dx=

∫

2

x(x−4)

dxcan be

integrated with a partial fraction

decomposition. Since

A

x

+

B

x− 4

=

2

x(x−4)

,

A=

− 1

2

andB=

1

2

. Therefore,