91Ó°ÊÓ

Evaluate the given double integral for the specified region \(R\).\(\iint_{R} 12 x^{2} e^{y^{2}} d A\), where \(R\) is the region in the first quadrant bounded by \(y=x^{3}\) and \(y=x\).

Sketch the region of integration for the given integral and set up an equivalent integral with the order of integration reversed.\(\int_{0}^{2} \int_{0}^{4-x^{2}} f(x, y) d y d x\)

Sketch the region of integration for the given integral and set up an equivalent integral with the order of integration reversed.\(\int_{0}^{1} \int_{0}^{2 y} f(x, y) d x d y\)

Sketch the region of integration for the given integral and set up an equivalent integral with the order of integration reversed.\(\int_{0}^{4} \int_{y / 2}^{\sqrt{y}} f(x, y) d x d y\)

Sketch the region of integration for the given integral and set up an equivalent integral with the order of integration reversed.\(\int_{1}^{e^{2}} \int_{\ln x}^{2} f(x, y) d y d x\)

Sketch the region of integration for the given integral and set up an equivalent integral with the order of integration reversed.\(\int_{0}^{\ln 3} \int_{e^{x}}^{3} f(x, y) d y d x\)

Sketch the region of integration for the given integral and set up an equivalent integral with the order of integration reversed.\(\int_{-1}^{1} \int_{x^{2}+1}^{2} f(x, y) d y d x\)

Sketch the region of integration for the given integral and set up an equivalent integral with the order of integration reversed.\(\int_{-1}^{1} \int_{-\sqrt{y+1}}^{\sqrt{y+1}} f(x, y) d y d x\)

Use a double integral to find the area of \(R\).\(R\) is the triangle with vertices \((0,-1),(-2,1)\), and \((2,1)\).

Use a double integral to find the area of \(R\).\(R\) is the region bounded by \(y=\frac{1}{2} x^{2}\) and \(y=2 x\).