Chapters

Exercise 1
Exercise 2
Exercise 3
Exercise 4
Exercise 5
Exercise 6
Exercise 7
Solution of exercise 1
Solution of exercise 2
Solution of exercise 3
Solution of exercise 4
Solution of exercise 5
Solution of exercise 6
Solution of exercise 7

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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.

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

$\text{[math]}$

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.

$\text{[math]}$

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.

$\text{[math]}$ $\text{[math]}$

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:

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$ $\text{[math]}$

Solution of Exercise 4

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

$\text{[math]}$ $\text{[math]}$

Solution of exercise 5

Find the volume generated by rotating the region bounded by $\text{[math]}$ and x = 2, around the y-axis.

As it turns about the y-axis, apply:

$\text{[math]}$

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.

$\text{[math]}$

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

$\text{[math]}$ $\text{[math]}$

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.

$\text{[math]}$ $\text{[math]}$ $\text{[math]}$ $\text{[math]}$ $\text{[math]}$ $\text{[math]}$ $\text{[math]}$ $\text{[math]}$

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Volume of Function Word Problems

Exercise 1

Exercise 2

Exercise 3

Exercise 4

Exercise 5

Exercise 6

Exercise 7

Solution of exercise 1

Solution of exercise 2

Solution of exercise 3

Solution of exercise 4

Solution of exercise 5

Solution of exercise 6

Solution of exercise 7

Theory

Problems of trigonometric integrals IV

Fundamental Theorem of Calculus II

Definite Integral

Trigonometric Integrals

Integral of a Power

Area of a Function

Area between Two Functions

Integration of Rational Functions

Integration Techniques Worksheet

Inverse Trigonometric Result

Table of Integrals

Average Function Value

Integration

Integration by Parts

Integration by Substitution

Fundamental Theorem of Calculus I

Trigonometric Integrals

Integral of Exponential Function

Volume of a Function

Formulas

Integral Formulas

Exercises

Solved Integrals Worksheet

Volume of Function Word Problems

Integral of a power Worksheet

Area of Function Problems

Problems of trigonometric integrals I

Problems Integrals Inverse Trigonometric Result

Integration Methods Worksheet

Exponential integral

Integration Worksheets

Integrals Worksheet

Definite Integral Worksheets

Integration Problems

Problems of trigonometric integrals II

Integral Worksheets

Problems of trigonometric integrals III

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