Skip to Content

Calculus 3: Double integrals

Set up a generic integral for the region bounded by the curves y=4xy = 4x and y=x3y = x^3, using the order of iteration dy/dxdy/dx.

Given an iterated integral with a function having y5+1y^5 + 1 in the denominator, reverse the order of integration to simplify the integral.

Find the volume of the solid bounded by the surfaces z=y+1z = y + 1 and z=x2+1z = x^2 + 1 over the region where y=x2y = x^2 and y=1y = 1.

Find the area bounded by the curves y=x2y = x^2 and x=4x = 4 using the double integral technique.

Solve a double integral problem using the Fundamental Theorem of Calculus.

Using the double integral method, find the volume of the given surface projected onto the xy-plane over a specified rectangular region.

Compute the volume under the surface given by f(x,y)=9x2y2f(x, y) = 9 - x^2 - y^2 over the rectangular region where xx is between 2-2 and 22 and yy is between 2-2 and 22.

Set up a generic integral for the region bounded by the curves y=4xy = 4x and y=x3y = x^3, using the order of iteration dy/dxdy/dx.

Given an iterated integral with a function having y5+1y^5 + 1 in the denominator, reverse the order of integration to simplify the integral.

Find the volume of the solid bounded by the surfaces z=y+1z = y + 1 and z=x2+1z = x^2 + 1 over the region where y=x2y = x^2 and y=1y = 1.

Find the area bounded by the curves y=x2y = x^2 and x=4x = 4 using the double integral technique.

Solve a double integral problem using the Fundamental Theorem of Calculus.

Find the volume under the surface f(x,y)=1+4xyf(x, y) = 1 + 4xy where xx ranges from 0 to 1 and yy ranges from 1 to 3.

Find the volume under the surface f(x,y)=3x23xy2f(x, y) = 3x^{2} - 3xy^{2} over the region bounded by y=x2y = x^{2} and y=2xy = 2x.

Solve a double integral problem involving a function of two variables over a specified domain.

Find the volume under the surface z=9x2y2z = 9 - x^2 - y^2 and above the xyxy-plane.

Find the surface area of the part of the function z=xyz = xy that lies inside the circle x2+y2=1x^2 + y^2 = 1 using double integrals.

Evaluate the integral 0π3sin(x)cos(π3x)dx\int_{0}^{\frac{\pi}{3}} \sin(x) \cos\left(\frac{\pi}{3} - x\right) \, dx.