Trivium
Given a region \(R\) in \(\mathbf{R}^2\) with a boundary that can be described analytically either in rectangular or polar coordinates, and a function \(f\colon R \to \mathbf{R},\) compute the value of \[ \iint\limits_R f \,\mathrm{d}A\,. \]
Given an iterated integral \(\iint_R f \,\mathrm{d}A\) of a function \(f\colon R \to \mathbf{R}\) on a region \(R \subset \mathbf{R}^2\) and a \(C^1,\) one-to-one transformation \(T\) of \(\mathbf{R}^2,\) compute the Jacobian of the transformation \(\left|\operatorname{J}_T\right|\) and write down the equivalent iterated integral over the pre-transformed domain, \[ \iint\limits_{T^{-1}(R)} \big(f \circ T\big) \,\bigl|\operatorname{J}_{T}\bigr| \,\mathrm{d}A\,. \]
Exercises
-
Compute the values of the following integrals on the indicated rectangles.
\(\displaystyle \iint\limits_R x-3y^2 \,\mathrm{d}A \\\text{for } R = \{(x,y) \mid 0\leq x\leq 2, 1\leq y\leq 2\}\)\(\displaystyle \iint\limits_R y\sin(xy) \,\mathrm{d}A \\\text{for } R = [1,2]\times[0,\pi]\)\(\displaystyle \iint\limits_R 7x^3y \,\mathrm{d}A \quad\text{for } R = [1,3]\times[-1,2]\)\(\displaystyle \iint\limits_R 2x-4y^3 \,\mathrm{d}A \quad\text{for } R = [-5,4]\times[0,3]\)\(\displaystyle \iint\limits_R y\mathrm{e}^{xy} \,\mathrm{d}A \quad\text{for } R = [1,2]\times[1,10]\)\(\displaystyle \iint\limits_R \frac{1}{(x+2y)^2} \,\mathrm{d}A \quad\text{for } R = [1,2]\times[2,5]\)
- For \(R = \{(x,y) \mid -1\leq x \leq 1, -2\leq y\leq 2\}\) evaluate the integral \(\iint_R \sqrt{1-x^2} \,\mathrm{d}A.\)
- Compute the volume of the solid \(S\) that is bounded by the elliptic paraboloid \(x^2+2y^2+z=16,\) the planes \(x=2\) and \(y=2,\) and the coordinate planes.
- Evaluate \(\iint_R x+2y \,\mathrm{d}A\) where \(R\) is the region in the \(xy\)-plane bounded by the parabolas \(y=2x^2\) and \(y=1+x^2.\)
- Compute the volume of the solid that lies under the paraboloid \(z = x^2+y^2\) and above the region \(R\) in the \(xy\)-plane bounded by the line \(y=2x\) and the parabola \(y=x^2.\)
- Compute the volume of the tetrahedron bounded by the planes \(x+2y+z = 2\) and \(x=2y\) and \(x=0\) and \(z=0.\)
- Evaluate the iterated integral \(\int_0^1\int_x^1 \sin\big(y^2\big) \,\mathrm{d}y\,\mathrm{d}x.\)
- Evaluate \(\iint_R 3x+4y^2 \,\mathrm{d}A\) where \(R\) is the region in the upper half-plane bound by the circles \(x^2+y^2=1\) and \(x^2+y^2=4.\)
- Evaluate the double integral \(\int_{-1}^{1}\int_{0}^{\sqrt{1-x^2}} x^2+y^2 \,\mathrm{d}y\,\mathrm{d}x\,.\)
- Calculate the volume of the solid that lies under the paraboloid \(z=x^2+y^2,\) above the \(xy\)-plane, inside the cylinder \(x^2+y^2=2x.\)
- Calculate the mass and the center of mass of the triangular lamina with vertices located at \((0,0)\) and \((1,0)\) and \((0,2)\) in the \(xy\)-plane having a density function \(\rho(x,y) = 1+3x+y.\)
- On a semi-circular lamina, suppose the density at any point is proportional to the distance from that point to the circle’s center. What are the coordinates of the lamina’s center of mass?
- Calculate the moments of inertia about the \(x\)-axis, about the \(y\)-axis, and about the origin, of a unit disk centered at the origin with radius \(r\) and uniform density \(\rho.\)
- Calculate the surface area of the portion of the surface \(z=x^2+2y+2\) that lies above the triangular region in the \(xy\)-plane with vertices \((0,0)\) and \((1,0)\) and \((1,1).\)
- Calculate the surface area of the portion of the paraboloid \(z=x^2+y^2\) that lies under the plane \(z=9.\)
- A transformation \(\mathbf{R}^2 \to \mathbf{R}^2\) is defined by the equations \({x=u^2-v^2}\) and \({y=2uv.}\) Sketch a picture of the image of the unit square \({(u,v) \in [0,1]\times[0,1]}\) under this transformation.
- Use the change of coordinates \({x=u^2-v^2}\) and \({y=2uv}\) to evaluate the integral \(\iint_R y \,\mathrm{d}A\) over the region \(R\) bounded by the \(x\)-axis and the parabolas \(y^2=4-4x\) and \(y^2=4+4x\) for \(y \geq 0.\)
-
Evaluate the following integral over the trapezoidal region \(R\)
with vertices \((1,0)\) and \((2,0)\) and \((0,-2)\) and \((0,-1).\)
\(\displaystyle \iint\limits_R \mathrm{e}^{(x+y)/(x-y)}\,\mathrm{d}A \)
Problems & Challenges
-
Consider the surface that is the graph of the function \(f(x,y) = \cos(x)\cos(y)\).
This surface, along with the \(xy\)-plane,
bound infinitely many congruent copies of a ravioli-shaped solid;
imagine the sheet of stuffed pasta before it is cut into individual raviolis.
- What is the volume contained within a single ravioli?
- Suppose that each ravioli is stuffed with a ricotta filling with density \(\rho = 1-z\); the ricotta has settled, becoming denser closer to the ravioli’s base. What is the mass of the ricotta in one ravioli?
- How high above the \(xy\)-plane is the center of mass of one ricotta-filled ravioli?
- What if instead of square raviolis, we decide to make circular ones? Consider the graph of \(g(x,y) = \cos\big(x^2+y^2\big)\) and the \(xy\)-plane as our two sheets of pasta. Except for a single ravioli centered at the origin bound between these two sheets, the rest of the pasta is a huge mess! This round ravioli experiment has been a disaster … oh well. What is the volume contained within this single ravioli? If we stuff this ravioli with a ricotta filling with density \(\rho = \frac{1}{z^2+1}\) how much ricotta do we need to fill the ravioli? How high above the \(xy\)-plane is the center of mass of this ricotta-filled ravioli?
- James Stewart Consider the region \(R = \big\{(x,y) \mid 1\leq x\leq 3, 2\leq y \leq 5\big\}.\) If \(\lfloor x \rfloor\) denotes the greatest integer in \(x,\) what must the value of the following integral be? \[ \iint\limits_R \lfloor x+y\rfloor \,\mathrm{d}A \]
- James Stewart Let \(\max\bigl(x^2, y^2\bigr)\) denote the larger of the two numbers \(x^2\) and \(y^2.\) Evaluate the integral \[ \int\limits_{0}^{1} \int\limits_{0}^{1} \mathrm{e}^{\max\bigl(x^2, y^2\bigr)}\,\mathrm{d}y\,\mathrm{d}x \]
- James Stewart Find the average value of the function \(f(x) = \int_x^1 \cos\bigl(t^2\bigr)\,\mathrm{d}t\) on the interval \([0,1].\)
- James Stewart The double integral \[ \int\limits_{0}^{1} \int\limits_{0}^{1} \frac{1}{1-xy} \,\mathrm{d}x\,\mathrm{d}y \] is an improper integral and could be defined as the limit of double integrals over the rectangle \([0,t] \times [0,t]\) as \(t\to 1^{-}.\) But if we expand the integrand as a geometric series, we can express the integral as the sum of an infinite series. Show that \[ \int\limits_{0}^{1} \int\limits_{0}^{1} \frac{1}{1-xy} \,\mathrm{d}x\,\mathrm{d}y = \sum_{n=1}^{\infty} \frac{1}{n^2} \]
-
James Stewart Leonhard Euler was able to find the exact sum of the series of the reciprocals of square integers. In 1736 he proved that \[\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}.\] Prove this fact by evaluating the double integral in the previous problem. Start by making the change of variables \[ x \to \frac{u-v}{\sqrt{2}} \qquad y \to \frac{u+v}{\sqrt{2}} \] This gives a rotation about the origin through the angle \(\pi/4.\) You will need to sketch the corresponding region in the uv-plane.