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3 votes
Let R be the region bounded by

y
=
7
sin
(
π
2
x
)
,
y
=
7
(
x

2
)
2
, and
y
=
x
+
6
, and containing the point (2,7).

Let R be the region bounded by y = 7 sin ( π 2 x ) , y = 7 ( x − 2 ) 2 , and y = x-example-1

1 Answer

5 votes

a. The area of
R is given by the integral


\displaystyle \int_1^2 (x + 6) - 7\sin\left(\frac{\pi x}2\right) \, dx + \int_2^(22/7) (x+6) - 7(x-2)^2 \, dx \approx 9.36

b. Use the shell method. Revolving
R about the
x-axis generates shells with height
h=x+6-7\sin\left(\frac{\pi x}2\right) when
1\le x\le 2, and
h=x+6-7(x-2)^2 when
2\le x\le\frac{22}7. With radius
r=x, each shell of thickness
\Delta x contributes a volume of
2\pi r h \Delta x, so that as the number of shells gets larger and their thickness gets smaller, the total sum of their volumes converges to the definite integral


\displaystyle 2\pi \int_1^2 x \left((x + 6) - 7\sin\left(\frac{\pi x}2\right)\right) \, dx + 2\pi \int_2^(22/7) x\left((x+6) - 7(x-2)^2\right) \, dx \approx 129.56

c. Use the washer method. Revolving
R about the
y-axis generates washers with outer radius
r_(\rm out) = x+6, and inner radius
r_(\rm in)=7\sin\left(\frac{\pi x}2\right) if
1\le x\le2 or
r_(\rm in) = 7(x-2)^2 if
2\le x\le\frac{22}7. With thickness
\Delta x, each washer has volume
\pi (r_(\rm out)^2 - r_(\rm in)^2) \Delta x. As more and thinner washers get involved, the total volume converges to


\displaystyle \pi \int_1^2 (x+6)^2 - \left(7\sin\left(\frac{\pi x}2\right)\right)^2 \, dx + \pi \int_2^(22/7) (x+6)^2 - \left(7(x-2)^2\right)^2 \, dx \approx 304.16

d. The side length of each square cross section is
s=x+6 - 7\sin\left(\frac{\pi x}2\right) when
1\le x\le2, and
s=x+6-7(x-2)^2 when
2\le x\le\frac{22}7. With thickness
\Delta x, each cross section contributes a volume of
s^2 \Delta x. More and thinner sections lead to a total volume of


\displaystyle \int_1^2 \left(x+6-7\sin\left(\frac{\pi x}2\right)\right)^2 \, dx + \int_2^(22/7) \left(x+6-7(x-2)^2\right) ^2\, dx \approx 56.70

Let R be the region bounded by y = 7 sin ( π 2 x ) , y = 7 ( x − 2 ) 2 , and y = x-example-1
User Darrel
by
7.5k points

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