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7E: Chapter Review Exercises

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Exercise $$\PageIndex{1}$$

Sketch the regions and evaluate the integrals:

1) $$\int_1^{\ln 8} \int_0^{\ln y} 2e^{x+y} \, dy dx$$

2) $$\int_0^{2} \int_x^{2} 3y^2 sin({xy}) \, dy dx$$

3) $$\int_0^{1} \int_0^{3} \frac{4x^2}{(y-1)^{2/3}} \, dy dx$$

4) $$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \frac{3}{(x^2+1)(y^2+1)} \, dy dx$$

5) $$\int \int_{x^2+y^2 \leq 1} \ln (x^2+y^2)\, dA$$

6) $$\int_0^2 \int_{y/2}^1 y e^{x^3} \,dx dy$$

7) $$\int \int_Q \frac{dA}{(1+x^2)(1+y^2)}$$, where $$Q$$ is the first quadrant of the $$xy-$$plane.

8) $$\int \int_R x \cos (y) \,dA$$, where $$R$$ is the region bounded by the coordinate axes and the curve $$y=1-x^2.$$