Signed Iterated Integral with Reversed Limits

Example

Evaluate

∫01∫xx2xy dy dx.\int_0^1\int_x^{x^2}xy\,dy\,dx.

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Rectangle and Fubini's Theorem

Example

Evaluate

∬R(x−3y2) dA,R=[0,2]×[1,2].\iint_R (x-3y^2)\,dA, \qquad R=[0,2]\times[1,2].

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Changing the Order of Integration

Example

Rewrite and evaluate

∫02∫x/21(x+2y) dy dx\int_0^2\int_{x/2}^{1}(x+2y)\,dy\,dx

by reversing the order.

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Polar Double Integral

Example

Evaluate

∬Dex2+y2 dA\iint_D e^{x^2+y^2}\,dA

where DD is the right half of the unit disk x2+y2≤1x^2+y^2\le1, x≥0x\ge0.

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Mass and Centroid of a Lamina

Example

A triangular lamina has vertices (0,0)(0,0), (1,0)(1,0), and (0,2)(0,2) and density

ρ(x,y)=1+3x+y.\rho(x,y)=1+3x+y.

Find its mass and center of mass.

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Planar Moment of Inertia

Example

A lamina of constant density ρ=3\rho=3 occupies the disk x2+y2≤4x^2+y^2\le4. Find its polar moment of inertia about the origin.

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Introductory Surface Area

Example

Find the area of the portion of the plane z=4−2x−yz=4-2x-y above the triangle in the xyxy-plane with vertices (0,0)(0,0), (2,0)(2,0), and (0,4)(0,4).

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Parametric Surface Area of a Cylinder

Example

Find the lateral surface area of the cylinder x2+y2=4x^2+y^2=4 for 0≤z≤30\le z\le3 using the parameterization

r(θ,z)=⟨2cos⁡θ,2sin⁡θ,z⟩,0≤θ≤2π,0≤z≤3.\mathbf{r}(\theta,z) = \langle 2\cos\theta,2\sin\theta,z\rangle, \qquad 0\le\theta\le2\pi,\quad 0\le z\le3.

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