Math 344: Calculus III |
1. Calculuate the iterated integral
\[ \int_0^1 \int_0^x \cos(x^2)\, dy\, dx. \]
2. Calculuate the iterated integral
\[ \int_0^1 \int_0^y \cos(x^2)\, dx\, dy. \]
3. Calculate the value of the double integral
\[ \int\!\!\!\!\int_D \dfrac{y}{1 + x^2}\, dA, \]
where \(D\) is bounded by \(y = \sqrt{x}\), \(y = 0\), \(x = 1\).
4. Describe the region whose area is given by the integral
\[ \int_0^{\frac{\pi}{2}} \int_0^{\sin 2\theta} r\, dr\, d\theta.\]
5. Describe the solid whose volume is given by the integral
\[ \int_0^{\frac{\pi}{2}} \int_0^{\frac{\pi}{2}} \int_1^2
\rho^2\sin\varphi\, d\rho\, d\varphi\, d\theta.\]
6. Calculate the value of the triple integral
\[ \int\!\!\!\!\int\!\!\!\!\int_H z^3\sqrt{x^2 + y^2 + z^2}\, dV \]
where \(H\) is the solid hemisphere that lies above the \(xy\)-plane
and has center \((0,0,0)\) and radius 1.
7. Give 5 other iterated integrals that are equivalent to
\[ \int_0^2 \int_0^{y^3} \int_0^{y^2} f(x,y,z)\, dz\, dx\, dy. \]
8. Use the transformation \( u = x - y\), \(v = x + y\) to
evaluate the integral
\[ \int\!\!\!\!\int_R \frac{x-y}{x+y}\, dA \]
where \(R\) is the square with vertices \((0,2)\), \((1,1)\), \((2,2)\),
and \((1,3)\).
9. Use the transformation \(x = u^2\), \(y = v^2\), and \(z = w^2\) to
find the volume of the region bounded by the surface \(\sqrt{x} + \sqrt{y} +
\sqrt{z} = 1\) and the coordinate planes.
10. Find the volume of the solid bounded above by the cone
\(z = \sqrt{x^2 + y^2}\) and below by the paraboloid \(z = x^2 + y^2\).
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