Exercise 1

Find the volume of the solid obtained by rotating the region bounded by y = sen x, x = 0 and x = π and the x-axis about the x-axis.

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Exercise 2

Calculate the volume of the cylinder generated by the rectangle bounded by straight lines y = 2, x = 1, x = 4, and the x-axis to rotate around the x-axis.

Exercise 3

Find the volume of the truncated cone generated by the trapezoid contained by the x-axis, the lines y = x + 2, x = 4 and x = 10, which turns around the x-axis.

Exercise 4

Find the volume generated by rotating the region bounded by y = 2x − x² and y = −x + 2 around the x-axis.

Exercise 5

Find the volume generated by rotating the region bounded by y²/8 = x and x = 2, around the y-axis.

Exercise 6

Calculate the volume of a sphere of radius r.

Exercise 7

Find the volume of the ellipsoid generated by the ellipse 16x² + 25y² = 400 and turning:

1 Around its major axis.

2 Around its minor axis.

 

 

Solution of exercise 1

Find the volume of the solid obtained by rotating the region bounded by y = sen x, x = 0 and x = π and the x-axis about the x-axis.

y = sen xx = 0x = π

V = \pi \int ^ {\pi} _ 0 sen^2 x dx = \pi \int ^{\pi} _0 ([\frac{1}{2} (1 - cos 2x)]) dx = \frac{\pi}{2} [x - \frac{1}{2} sen2x]^{\pi}_{0} = \frac{\pi^2}{2} u^3

 

Solution of exercise 2

Calculate the volume of the cylinder generated by the rectangle bounded by straight lines y = 2, x = 1, x = 4, and the x-axis to rotate around the x-axis.

V = \pi \int ^{4}_{1} 2^2 dx = 4\pi [x]^{4}_{1} = 4\pi(4 - 1) = 12\pi u^3

 

Solution of exercise 3

Find the volume of the truncated cone generated by the trapezoid contained by the x-axis, the lines y = x + 2, x = 4 and x = 10, which turns around the x-axis.

V = \int^{10}_{4} (x + 2)^2 dxV = \pi \int^{10}_{4} (x^2 + 4x + 4)^2 dx = \pi [\frac{x^3}{3} + 2x^2 + 4x]^{10}_{4}

= \pi (\frac{1000}{3} + 200 + 40 - \frac{64}[3} - 32 - 16) = 504 \pi u^3

 

Solution of exercise 4

Find the volume generated by rotating the region bounded by y = 2x − x² and y = −x + 2 around the x-axis.

The points of intersection between the parabola and the line:

\systeme{y = 2x - x^2, y = -x + 2}

2x - x^2 = -x + 2

(1,1)

(2,0)

Solution of Exercise 4

The parable is above the line in the interval of integration.

V = \pi \int ^{2}_{1} [ (2x - x^2)^2 - (-x + 2)^2] dx = \pi \int ^{2}_{1} (x^4 - 4x^3 + 3x^2 + 4x - 4) dx

= \pi [\frac{1}{5} x^5 - x^4 + x^3 + 2x^2 - 4x]^{2}_{1} = \frac{\pi}{5}u^3

 

Solution of exercise 5

Find the volume generated by rotating the region bounded by \frac{y^2}{8} = xand x = 2, around the y-axis.

As it turns about the y-axis, apply:

V = \pi \int ^{b}_{a} x^2 dy

Solution of Exercise 5

Since the parabola is symmetrical about the x-axis, the volume is equal to two times the volume generated between y = 0 and y = 4.

V = 2 \pi \int ^{4}_{0} 2^2 dy - 2\pi \int ^{4}_{0} (\frac{y^2}{8})^2 dy = 2 \pi[4y - \frac{y^5}{320}]^{4}_{0} = \frac{128}{5} \pi u^3

 

Solution of exercise 6

Calculate the volume of a sphere of radius r.

Start from the equation of the circumference x² + y² = r².

Turning a semicircle around the x-axis gives a sphere.

Solution of Exercise 6

V = \pi \int ^{r}_{-r} (\sqrt {r^2 - x^2}) dx = \pi \int ^{r}_{-r} (r^2 - x^2) dx

= \pi [r^2 x - \frac{x^3}{3}]^{r}_{-r} = \pi (\frac{2r^3}{3} + \frac{2r^3}{3}) = \frac{4}{3} \pi r^3

 

Solution of exercise 7

Find the volume of the ellipsoid generated by the ellipse 16x² + 25y² = 400 and turning:

1 Around its major axis.

2 Around its minor axis.

Solution of Exercise 7

As the ellipse is symmetric about two axes, the volume is double the portion generated by the ellipse in the first quadrant in both cases.

16x^2 + 25y^2 = 400

y^2 = \frac{400 - 16x^2}{25}

(5, 0)

V_A = 2\pi \int^{5}_{0} (\frac{400 - 16x^2}{25}) dx = 2\pi[16x - \frac{16}{75}x^3] ^{5}_{0} = \frac{320}{3} \pi u^3

16x^2 + 25y^2 = 400

x^2 = \frac{400 - 25y^2}{16}

(0, 4)

V_2 = 2\pi \int^{4}_{0} (\frac{400 - 25y^2}{16}) dy = 2\pi [25y - \frac{25}{48}y^3]^{4}_{0} = \frac{400}{3} \pi u^3

 

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.