Let f and g be the functions given by f(x)=1+sin(2x) and g(x)=e^(x/2). Let R be the shaded region in the first quadrant enclosed by the graphs of f and g.
A. The region R is the base of a solid. For this solid, the cross sections, perpendicular to the spx-axis, are rectangles with height 5. Find the volume of this solid.
B. The region R is the base of a solid. For this solid, the cross sections, perpendicular to the x-axis, are squares. Find the volume of this solid.
the graphs intersect at (0,1) and (1.136,1.764)
A
The cross-sections have base of f(x)-g(x) and height 5. So, we add them all up:
v = ∫[0,1.136] 5(1+sin(2x)-e^(x/2)) dx
= 5(x - 1/2 cos(2x) - 2e^(x/2)) [0,1.136]
= 2.1455
B
The same thing, but the height is the same as the base, so
v = ∫[0,1.136] ((1+sin(2x)-e^(x/2))^2 dx
you'll have to use integration by parts to do the sin(2x)*e^(x/2) stuff.
If you scroll down a bit here, you can see the final indefinite integral:
http://www.wolframalpha.com/input/?i=%E2%88%AB%5B0%2C1.136%5D+%28%281%2Bsin%282x%29-e%5E%28x%2F2%29%29%5E2+dx
Here was my first answer
http://www.jiskha.com/display.cgi?id=1429478661
In a subsequent reply you said you were confused.
Now you simply repost the same question for a third time. Exactly what did you not understand in my answer?
I replied to those ones again
To find the volumes of the solids, we need to determine the limits of integration for the region R.
For part A, where the cross sections are rectangles with height 5:
1. We first need to find the x-values where the graphs of f(x) and g(x) intersect. Setting f(x) equal to g(x), we have:
1 + sin(2x) = e^(x/2)
2. To solve this equation, we can use numerical or graphical methods. One possible approximation is to use a graphing calculator to find the x-intercepts of the functions.
3. Once we have the x-values of the intersection points, we can set up the integral for the volume of the solid. Since the cross-sectional rectangles have a fixed height of 5, the width of each rectangle will be the difference between the y-values of f(x) and g(x) at each x-value.
4. The integral representing the volume of the solid is then given by:
V = ∫[a,b] (f(x) - g(x)) dx
5. Evaluating this integral will give us the volume of the solid.
For part B, where the cross sections are squares:
1. We need to determine the x-values where the graphs of f(x) and g(x) intersect, as in part A.
2. Once we have the x-intercepts, we need to find the corresponding y-values on both curves.
3. The width of each square cross section is equal to the difference between the x-values at each intersection point.
4. The integral representing the volume of the solid is given by:
V = ∫[a,b] (f(x) - g(x))^2 dx
5. Evaluating this integral will give us the volume of the solid.
Please note that the specific values for the integration limits (a and b) will depend on the intersection points and the region of interest within the first quadrant.