Chapter 7

Evaluate the double integrals in Exercises 1 through 18 .\(\int_{0}^{1} \int_{1}^{2} x^{2} y d x d y\)

Evaluate the double integrals.\(\int_{1}^{2} \int_{0}^{1} x^{2} y d y d x\)

V\(\int_{0}^{\ln 2} \int_{-1}^{0} 2 x e^{y} d x d y\)

Evaluate the double integrals.\(\int_{2}^{3} \int_{-1}^{1}(x+2 y) d y d x\)

Evaluate the double integrals.\(\int_{1}^{3} \int_{0}^{1} \frac{2 x y}{x^{2}+1} d x d y\)

Evaluate the double integrals.\(\int_{0}^{4} \int_{-1}^{1} x^{2} y d y d x\)

Evaluate the double integrals.\(\int_{0}^{1} \int_{1}^{5} y \sqrt{1-y^{2}} d x d y\)

Evaluate the double integrals.\(\int_{2}^{3} \int_{1}^{2} \frac{x+y}{x y} d y d x\)

Evaluate the double integrals.\(\int_{1}^{2} \int_{2}^{3}\left(\frac{y}{x}+\frac{x}{y}\right) d y d x\)

Evaluate the double integrals.\(\int_{0}^{4} \int_{0}^{\sqrt{x}} x^{2} y d y d x\)

Evaluate the double integrals.\(\int_{0}^{1} \int_{1}^{5} x y \sqrt{1-y^{2}} d x d y\)

Evaluate the double integrals.\(\int_{0}^{1} \int_{y-1}^{1-y}(2 x+y) d x d y\)

Evaluate the double integrals.\(\int_{0}^{1} \int_{0}^{4} \sqrt{x y} d y d x\)

Evaluate the double integrals.\(\int_{0}^{1} \int_{x}^{2 x} e^{y-x} d y d x\)

Evaluate the double integrals.\(\int_{1}^{e} \int_{0}^{\ln x} x y d y d x\)

Evaluate the double integrals.\(\int_{0}^{3} \int_{y^{2} / 4}^{\sqrt{10-y^{2}}} x y d x d y\)

Use inequalities to describe \(R\) in terms of its vertical and horizontal cross sections.\(R\) is the region bounded by \(y=x^{2}\) and \(y=3 x\).

Use inequalities to describe \(R\) in terms of its vertical and horizontal cross sections.\(R\) is the region bounded by \(y=\sqrt{x}\) and \(y=x^{2}\).

Use inequalities to describe \(R\) in terms of its vertical and horizontal cross sections.\(R\) is the rectangle with vertices \((-1,1),(2,1)\), \((2,2)\), and \((-1,2)\).

Use inequalities to describe \(R\) in terms of its vertical and horizontal cross sections.\(R\) is the triangle with vertices \((1,0),(1,1)\), and \((2,0)\).

Use inequalities to describe \(R\) in terms of its vertical and horizontal cross sections.\(R\) is the region bounded by \(y=\ln x, y=0\), and \(x=e\).

Evaluate the given double integral for the specified region \(R\).\(\iint_{R} 3 x y^{2} d A\), where \(R\) is the rectangle bounded by the lines \(x=-1, x=2, y=-1\), and \(y=0\).

Evaluate the given double integral for the specified region \(R\).\(\iint_{R}(x+2 y) d A\), where \(R\) is the triangle with vertices \((0,0),(1,0)\), and \((0,2)\).

Evaluate the given double integral for the specified region \(R\).\(\iint_{R} 48 x y d A\), where \(R\) is the region bounded by \(y=x^{3}\) and \(y=\sqrt{x}\).

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

Evaluate the given double integral for the specified region \(R\).\(\iint_{R} 12 x d A\), where \(R\) is the region bounded by \(y=x^{2}\) and \(y=6-x\)

Evaluate the given double integral for the specified region \(R\).\(\iint_{R}(2 x+1) d A\), where \(R\) is the triangle with vertices \((-1,0),(1,0)\), and \((0,1)\).

Evaluate the given double integral for the specified region \(R\).\(\iint_{R} 2 x d A\), where \(R\) is the region bounded by \(y=\frac{1}{x^{2}}, y=x\), and \(x=2\).

Evaluate the given double integral for the specified region \(R\).\(\iint_{R} \frac{1}{y^{2}+1} d A\), where \(R\) is the triangle bounded by the lines \(y=\frac{1}{2} x, y=-x\), and \(y=2\).

Evaluate the given double integral for the specified region \(R\).\(\iint_{R} e^{y^{3}} d A\), where \(R\) is the region bounded by \(y=\sqrt{x}, y=1\), and \(x=0\).

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 \(y=2 x\).

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

Use a double integral to find the area of \(R\).\(R\) is the region bounded by \(y=x^{2}-4 x+3\) and the \(x\) axis.

Use a double integral to find the area of \(R\).\(R\) is the region bounded by \(y=x^{2}+6 x+5\) and the \(x\) axis.

Use a double integral to find the area of \(R\).\(R\) is the region bounded by \(y=\ln x, y=0\), and \(x=e\).

Use a double integral to find the area of \(R\).\(R\) is the region in the first quadrant bounded by \(y=4-x^{2}, y=3 x\), and \(y=0\).

Use a double integral to find the area of \(R\).\(R\) is the region bounded by \(y=\frac{16}{x}, y=x\), and \(x=8\).

Find the volume of the solid under the surface \(z=f(x, y)\) and over the given region \(R\).\(f(x, y)=6-2 x-2 y\) \(R: 0 \leq x \leq 1,0 \leq y \leq 2\)

Find the volume of the solid under the surface \(z=f(x, y)\) and over the given region \(R\).\(f(x, y)=9-x^{2}-y^{2}\) \(R:-1 \leq x \leq 1,-2 \leq y \leq 2\)

Find the volume of the solid under the surface \(z=f(x, y)\) and over the given region \(R\).$$ f(x, y)=\frac{1}{x y} $$ \(R: 1 \leq x \leq 2,1 \leq y \leq 3\)

Find the volume of the solid under the surface \(z=f(x, y)\) and over the given region \(R\).$$ f(x, y)=e^{x+y} $$ \(R: 0 \leq x \leq 1,0 \leq y \leq \ln 2\)