Question 5.16

Evaluate each of the following integrals if \(R=[0,1]\times [0,1]\).

• (a) \(\displaystyle\int\!\!\!\int_{R}\, (x^3 +y^2)\ {\it dA}\)
• (b) \(\displaystyle\int\!\!\!\int_{R}\, ye^{xy} \,{\it dA}\)
• (c) \(\displaystyle\int\!\!\!\int_{R}\, (xy)^2 \cos x^3 \,{\it dA}\)
• (d) \(\displaystyle\int\!\!\!\int_{R}\, \ln\, [(x+1)(y+1)]\ {\it dA}\)

Question 5.17

Evaluate each of the following integrals if \(R=[0,1]\times [0,1].\)

• (a) \(\displaystyle\int\!\!\!\int_{R}\, (x^m y^n) \, {\it dx}\, {\it dy} \), where \(m,n>0\)
• (b) \(\displaystyle\int\!\!\!\int_{R} \, (ax +by +c)\, {\it dx}\, {\it dy}\)
• (c) \(\displaystyle\int\!\!\!\int_{R}\, \sin \ (x+y) \, {\it dx}\, {\it dy}\)
• (d) \(\displaystyle\int\!\!\!\int_{R}\, (x^2 +2xy +y \sqrt{x})\, {\it dx}\, {\it dy}\)

Question 5.18

Evaluate over the region \(R\): \[ {\intop\!\!\!\intop}_{R} \frac{yx^{3}}{y^{2}+2} \, {\it dy}\, {\it dx} , \quad \quad R{:}\ [0,2] \times [-1,1]. \]

Question 5.19

Evaluate over the region \(R\): \[ {\intop\!\!\!\intop}_{R} \frac{y}{1 + x^2} \, {\it dx}\, {\it dy} , \quad \quad R{:}\ [0,1] \times [-2,2]. \]

Question 5.20

Sketch the solid whose volume is given by: \[ \int_{0}^{1} \int_{0}^{1} (5 - x - y) \, {\it dy}\, {\it dx}. \]

Question 5.21

Sketch the solid whose volume is given by: \[ \int_{0}^{3} \int_{0}^{2} (9 + x^2 + y^2) \, {\it dx}\, {\it dy}. \]

Question 5.22

Compute the volume of the region over the rectangle \([0, 1]\times [0, 1]\) and under the graph of \(z=xy\).

Question 5.23

Compute the volume of the solid bounded by the \(xz\) plane, the \(yz\) plane, the \(xy\) plane, the planes \(x = 1\) and \(y= 1\), and the surface \(z= x^2 + y^4\).

Question 5.24

Let \(f\) be continuous on \([a, b]\) and \(g\) continuous on \([c,d]\). Show that \[ \intop\!\!\!\intop\nolimits_{R}\, [f(x)g(y)]\, {\it dx}\, {\it dy} = \bigg[\int^b_a f (x)\, {\it dx} \bigg] \bigg[\int_c^d g(y)\, {\it dy} \bigg], \] where \(R = [a,b] \times [c,d\,]\).

Question 5.25

Compute the volume of the solid bounded by the surface \(z=\sin y\), the planes \(x= 1, x = 0, y = 0\), and \(y= \pi/2\), and the \(xy\) plane.

Question 5.26

Compute the volume of the solid bounded by the graph \(z=x^2+y\), the rectangle \(R = [0, 1]\times [1, 2]\), and the “vertical sides” of \(R\).

Question 5.27

Let \(f\) be continuous on \(R = [a,b] \times [c,d]\); for \(a<x<b,c<y<d\), define \[ F(x,y)= \int^x_a \int^y_c f(u,v){\,d} v {\,d} u. \] Show that \(\partial^2 F / \partial x\, \partial y =\partial^2 F/\partial y\, \partial x= f(x,y)\). Use this example to discuss the relationship between Fubini’s theorem and the equality of mixed partial derivatives.

Question 5.28

Consider the integral in 2(a) as a function of \(m\) and \(n\); that is, \[ f(m,n) := {\intop\!\!\!\intop}_{R} x^m y^n \, {\it dx}\, {\it dy}. \] Evaluate \(\displaystyle\lim\nolimits_{m,n \rightarrow \infty} f(m,n)\).

Question 5.29

Let: \[ f(m,n) := \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \cos nx \sin my \, {\it dx}\, {\it dy}. \] Show that \(\lim_{m,n \rightarrow \infty} f(m,n) = 0\).

Question 5.30

Let \(f \colon\ [0,1] \times [0,1] \rightarrow {\mathbb R}\) be defined by \[ f(x,y)= \left \{\begin{array}{ll} 1 & x \hbox{ rational}\\[3pt] 2y & x \hbox{ irrational}. \end{array}\right. \] Show that the iterated integral \(\int^1_0 \big[\int^1_0 f(x,y)\, {\it dy} \big] {\it dx}\) exists but that \(f\) is not integrable.

Question 5.31

Express \({\intop\!\!\!\intop}_{R}\) cosh \({\it xy}\, {\it dx}\, {\it dy}\) as a convergent sequence, where \(R =[0, 1]\times [0, 1]\).

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Question 5.32

Although Fubini’s theorem holds for most functions met in practice, we must still exercise some caution. This exercise gives a function for which it fails. By using a substitution involving the tangent function, show that \[ \int^1_0\int^1_0\frac{x^2-y^2}{(x^2+y^2)^2}\,{\it dy}\,{\it dx}=\frac{\pi}{4}, \] yet \[ \int^1_0\int^1_0\frac{x^2-y^2}{(x^2+y^2)^2}\,{\it dx}\,{\it dy}=-\frac{\pi}{4}. \] Why does this not contradict Theorem 3 or 3\('\)?

Question 5.33

Let \(f\) be continuous, \(f \geq 0\), on the rectangle \(R\). If \({\intop\!\!\!\intop}_{R}f\ {\it dA}=0\), prove that \(f = 0\) on \(R\).