Area between the curves

Find the area between the curves.

x= -2, x= 1, y= 7x, y=x^2 -8

Set up the integral (or integrals)needed to compute this area. Use the smallest possible number of integrals.

For video explanation, click here.

U-Substitution: integrals

It Use substitution to find the indefinite integral.

int(3dm/(3m-7)^6)

For video explanation, click here.

Volumes - Cylindrical Shell

Use the shell method to find the volume generated by revolving the shaded region about the y-axis.

Volumes - Cross Section

A solid lies between planes perpendicular to the x-axis at x=0 and x=10. The cross-sections perpendicular to the x-axis on the interval 0 ≤ x ≤ 10 are squares with diagonals that run from the parabola y=-2sqrt(x) to the parabola y=2sqrt(x). Find the volume of the solid.

Volumes - Cross Section

Find the volume of the following solids. The base of a solid is the region between the curve y=2sqrt(sinx) and the interval ​[0,π​] on the​ x-axis. The​ cross-sections perpendicular to the​ x-axis are: a) Equilateral triangles with bases running from the x-axis to the curve as shown in the figure.

Volumes - Cross Section

Find the volume of the following solids. The base of a solid is the region between the curve y=2sqrt(sinx) and the interval ​[0,π​] on the​ x-axis. The​ cross-sections perpendicular to the​ x-axis are: b) Squares with bases running from the x-axis to the curve.

Volumes - Cross Section

A solid lies between planes perpendicular to the y-axis at y = 0 and y = 2. The cross-sections perpendicular to the y-axis are circular disks with diameters running from the y-axis to the parabola x=sqrt(14)*y^2. Find the volume of the solid.

Volumes - Cross Section

Find the volume of the given right tetrahedron. (Hint: Consider slices perpendicular to one of the labeled edges.)

Volumes - Cross Section

Find the volume of the solid generated by revolving the shaded region about the x-axis.

Volumes - Cross Section

Find the volume of the solid generated by revolving the shaded region about the y-axis.

Volumes - Cross Section

Find the volume of the solid generated by revolving the region bounded by the given line and curve about the x-axis. y=sqrt(9-x^2), y = 0.

Volumes - Cross Section

Find the volume of the solid generated by revolving the region bounded by the given curve and lines about the x-axis.

y= exp(x-4), y= 0 x= 4, x= 6

Volumes - Cross Section

Find the volume of the solid generated by revolving the region about the given line. The region in the first quadrant bounded above by the line y= sqrt(2), below by the curve y= csc x cot x, and on the right by the line x=π/2, about the line y= sqrt(2).

Volumes - Cross Section

Find the volume of the solid generated by revolving the region enclosed by x= sqrt(15)*y^2, x= 0, y= -1 and y= 1 about the y-axis.

Volumes - Cross Section

Use the washer method to find the volume of the solid generated by revolving the shaded region about the x-axis.

Volumes - Cross Section

Find the volume of the solid generated by revolving the region bounded by the given curve and lines about the x-axis. y= 5x, y= 5, x= 0.

Volumes - Cross Section

Find the volume of the solid generated by revolving the region enclosed by the triangle with vertices (3,3), (3,6), and (7,6) about the y-axis.

Volumes- Cylindrical Shell

Use the shell method to find the volume of the solid generated by revolving the shaded region about the x-axis.

Volumes - Cylindrical Shell

Use the shell method to find the volume of the solid generated by revolving the shaded region about the y-axis.

Volumes - Cylindrical Shell

Use the shell method to find the volume of the solid generated by revolving the region bounded by y= 4x, y= -x/2, and x= 3 about the y-axis.

Volumes - Cylindrical Shell

Use the shell method to find the volume of the solid generated by revolving the region bounded by y= 9x-8, y= sqrt(x), and x= 0 about the y-axis.

Volumes - Cylindrical Shell

a) Show that xf(x) = sin x, 0 ≤ x ≤ π

b) Find the volume of the solid generated by revolving the shaded region about the y-axis in the accompanying figure.

Volumes - Cylindrical Shell

Use the shell method to find the volume of the solid generated by revolving the region bounded by x= 6sqrt(y), x= -y, and y= 2 about the x-axis.

Volumes - Cylindrical Shell

Use the shell method to find the volume of the solid generated by revolving the regions bounded by the curves and lines about the x-axis.

y= |x|/2, y= 3

Volumes - Cylindrical Shell

Use the shell method to find the volume of the solid generated by revolving the regions bounded by the curves and lines about the x-axis.

y=sqrt(x), y= 0, y=(x-3)/2

Volumes - Cylindrical Shell

Use the shell method to find the volume of the solid generated by revolving the region bounded by the line y= x+2 and the parabola y= x^2 about the following lines. a) The line x= 2; b) The line x= -1; The x-axis; d) The line y=4

Volumes - Cylindrical Shell

Compute the volume of the solid generated by revolving the region bounded by y= 5x and y = x^2 about each coordinate axis using the methods below.

a) Using the shell method, set up the integral to find the volume of the solid generated by rotating the region bounded by y= x^2 and y=5x about the x-axis.

b)Using the shell method, set up the integral to find the volume of the solid generated by rotating the region bounded by y=x^2 and y= 5x about the y-axis.

Trigonometric Integrals

Evaluate the integral

int(tan(3x)sec^2(3x)dx

Trigonometric Integrals

Evaluate the integral

int(sec^6(5x)tan(5x)dx

Trigonometric Integrals

Evaluate the integral

int(sec^2(6x)tan^3(6x)dx

Trigonometric Integrals

Evaluate the integral

int(20csc^4(2θ)dθ

Trigonometric Integrals

Evaluate the integral

int(6tan^3(x)dx

Trig Substitutions

Evaluate the following integral. What substitution will be the most helpful for evaluating this​ integral? Find dx. Rewrite the given integral using this substitution. Evaluate the indefinite integral.

Trig Substitutions

Use Trigonometric Substitution to evaluate the following integral. Sketch​ a​ triangle, and label the sides of the triangle appropriately. Label​ the​ angle,​ θ. Label the right angle with a box.

Trig Substitutions

The integral in this exercise converges. Evaluate the integral without using a table.

Trig Substitutions

Evaluate the integral using integration by parts.

int(xe^(7x))dx

Integral Rational Function

The following integral converges. Evaluate the integral without using tables.

Integration By Parts

Evaluate the following integral. Use integration by parts to rewrite the integral.

Trig Substitutions

Evaluate the following integral. What substitution will be the most helpful for evaluating this​ integral? Find dx. Rewrite the given integral using this substitution. Evaluate the definite integral.

Integral Rational Function

Evaluate the following integral. Find the partial fraction decomposition of the integrand.

Improper Integral

For what values of p does ∫1∞x^(−p)dx ​converge?

Integral Rational Function

Expand the quotient by partial fractions.

(7x-12)/(x^2-3x+2)