Brain Gains (Rapid Recall, Jivin' Generation)

  • Consider the region in the first quadrant that lies left of the line $y=x/2$ and below the line $y=1$. A metal plate occupies this region, and has density given by $\delta = \cos(y^2)$. Set up an iterated integral of the form $\int_?^?\int_?^? ? dydx$ that would give the mass of the plate.
  • For the same region, set up an iterated integral of the form $\int_?^?\int_?^? ? dxdy$ that would give the mass of the plate.
  • Examine the two integrals above. One of them has a problem an cannot be computed. Which one, and why?
  • Set up the integrals to compute $\bar x$, the $x$-coordinate of the center of mass of the metal plate.

Mathematica Solutions

The first integral has a problem in that the inside integral cannot be done. Mathematica has been coded in a way that it can recognize this problem here, and still gives an exact answer for the double integral. Putting an N in front of integrate swaps from getting an exact answer to a numerical approximation. 

Integrate[Cos[y^2], {y, x/2, 1}]
Integrate[Cos[y^2], {x, 0, 2}, {y, x/2, 1}]
NIntegrate[Cos[y^2], {x, 0, 2}, {y, x/2, 1}]

The second integral can be done with substitution. 

Integrate[Cos[y^2], {x, 0, 2 y}]
Integrate[Cos[y^2], {y, 0, 1}, {x, 0, 2 y}]

The average value formulas require numerical integration, no matter how we go about doing things. Let's look at Mathematica's solution. 

Integrate[x Cos[y^2], {y, 0, 1}, {x, 0, 2 y}]/Integrate[Cos[y^2], {y, 0, 1}, {x, 0, 2 y}]
NIntegrate[x Cos[y^2], {y, 0, 1}, {x, 0, 2 y}]/NIntegrate[Cos[y^2], {y, 0, 1}, {x, 0, 2 y}]

Group Problems

  1. Consider the integral $\ds\int_{0}^{3}\int_{0}^{x}dydx$.
    • Shade the region whose area is given by this integral.
    • Compute the integral.
    • Now compute $\ds\int_{0}^{x}\int_{0}^{3}dxdy$. Do you get a single number or an expression involving a variable? What's wrong with this integral? How can you recognize there will be a problem without even computing a single integral?
    • Adjust the bounds on the previous integral (use the order $dxdy$) so that the bounds describe the same region as original integral, but make sure the bounds on $y$ are between two constants. In other words, fill in the question marks below so that the integral's bounds describe the same region as the first part of this problem. $$\ds\int_{?}^{?}\int_{?}^{?}dxdy.$$
  2. Set up an integral formula to compute each of the following:
    • The mass of a disc that lies inside the circle $x^2+y^2=9$ and has density function given by $\delta = x+10$
    • The $x$-coordinate of the center of mass (so $\bar x$) of the disc above.
    • The $z$-coordinate of the center-of-mass (so $\bar z$) of the solid object in the first octant (all variables positive) that lies under the plane $2x+3y+6z=6$ (so $\frac{x}{3}+\frac{y}{2}+\frac{z}{1}=1$).
    • The $y$-coordinate of the center-of-mass (so $\bar y$) of the same object.
  3. Consider $\int_{0}^{4}\int_{x}^{4}e^{y^2}dydx$.
    • Draw the region described by the bounds.
    • Swap the order of the bounds on the integral (use $dxdy$ instead of $dydx$).
    • Compute the integral.