10.1: Double Integrals over Rectangular Regions
- Page ID
- 144350
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- Compute \(\displaystyle \int_0^2 \int_0^4 (1 + x)\ dy\,dx\).
- Compute \(\displaystyle \int_{-1}^1 \int_{-2}^2 (2x + 3y + 5)\ dx\,dy\).
- Compute \(\displaystyle \int_1^6 \int_2^9 \dfrac{\sqrt{y}}{x^2}\ dy\,dx\).
- Compute \(\displaystyle \int_{\ln 2}^{\ln 3} \int_0^1 e^{x+y}\ dy\,dx\).
- Compute \(\displaystyle \int_0^{\pi} \int_0^{\pi/2} \sin(2x)\cos(3y)\ dx\,dy\).
- Compute \(\displaystyle \int_1^e \int_1^e \dfrac{\sin(\ln x)\cos(\ln y)}{xy}\ dx\,dy\).
- Compute \(\displaystyle \int_0^1 \int_1^2 \dfrac{x}{x^2 + y^2}\ dy\,dx\).
- Find the volume under the surface \(z = xy\) and over rectangle \(R = [0, 2] \times [1, 3]\).
- Find the volume under the surface \(z = y^2\) and over rectangle \(R = [-2, 2] \times [-1, 4]\).
- Find the volume under the surface \(z = \dfrac{x\sqrt{4-x^2}}{y}\) and over rectangle \(R = [0, 2] \times [1, e]\).
- Sketch the solid whose volume is expressed as \(\displaystyle \iint_R x\ dA\), where \(R = [0, 5] \times [0, 5]\). Then use geometry to find the volume.
- Sketch the solid whose volume is expressed as \(\displaystyle \iint_R \sqrt{4 - y^2}\ dA\), where \(R = [0, 2] \times [0, 2]\). Then use geometry to find the volume.