Stewart Calculus 8e 7.Plus Problem hints
- 1. cutting a circle into equal parts with parallel lines.
Just treat a unit circle (you can always scale up to the actual radius of
the pie at the end) with its standard equation in the x-y plane.
Maple can just as easily do the integral in terms of x but the
solution requires a numeric approximation.
If you are clever enough with your keyword choice to
find a
Google hit solution, you will notice that instead of using x and
y to
describe the circle, the polar angle substitution x = cos(θ) is
preferable to change the independent variable (for the unit circle case), which is a change of variable
in the definite integral you set up to find the unknown x value x =
a such
that the part of the unit circle to its right has one third the area of the unit
circle. Maple can do this integral easily to obtain a condition on the angle that
can be solved numerically. This derivation is an example of a trigonometric
substitution method which is how Maple gets the antiderivative to display,
and was necessary for students to do last century.
- 3. radical difference integral.
Maple easily integrates this but what to do with the special function
answer? Try approximating and then guess what the output means effectively.
To show this exactly, realize that this is the difference area between the
two curves which are the graphs of the two radical expressions. If you look
at the graph of these two radical expressions, symmetry will also tell you
the answer.
Alternatively re-express this difference area in terms of y as the
independent variable, realizing that y is a dummy variable, change back to
the dummy variable x and compare it to the original integral. What can you
conclude? [I would never have gotten this without a hint!] How are these two
functions related?
- 4. overlapping circles.
put one circle at the origin, the other with center on the positive x axis.
you have to find the area of the overlap and then figure out how to subtract
that from the sum of the areas appropriately to avoid overcounting it.
- 5. area between ellipse and circle
Use the standard equation for an ellipse and circle at the origin. Make a
diagram. Set up the difference area integral. Re-interpret it in terms of
the same calculation for the second ellipse.
- 11. limit.
Maple can do this. But what rule of limits does Maple use here? First do the
integral and check what happens at t = 0, then take its limit.
- 12. exp(sin).
Use Maple of course. To find the critcal point, you need to draw a diagram
of the upper half of the unit circle and determine the relationship between
points in the first and second quadrants which have the same sine value.
This condition can even be solved by hand.
- 15. circle in wedge.
Draw a diagram with the radius connecting the point of tangency to make a
right triangle. Draw a horizontal line to the y axis from this point.
Identify 3 similar right triangles in your diagram. Using similar triangle
ratios you can determine the location of the center of the circle on the y
axis. Once that is done, you can easily set up the difference area integral
and evaluate it. You can also use the matching slope condition between the
straight line and the circle at the point of tangency without knowing the y
value for the circle center in order to determine the value of x there.
There are often different paths to a solution.
- 16. rocket losing mass.