find the volume of the solid bounded above by the surface z=f(x,y) and below by the plane region R
f(x,y)=2x^2 y;R is the region bounded by the graphs of y=x and y=x^2
v = ∫∫R f(x,y) dA
= ∫[0,1]∫[x,x^2] 2x^2 y dy dx
= ∫[0,1] (x^2 y^2)[x,x^2] dx
= ∫[0,1] (x^6 - x^4) dx
= (1/7 x^7 - 1/5 x^5)[0,1]
= 1/7 - 1/5
= -2/35
Oops. That sign is reversed, since y f goes from x^2 to x.
Answer is 2/35
Couldn't figure how I got a minus value for a function that is positive over the region.
To find the volume of the solid bounded by the surface z=f(x,y) and the plane region R, you can use a double integral over the region R.
1. Start by finding the limits of integration for x and y.
- Since R is bounded by the graphs of y=x and y=x^2, we need to find the intersection points of these two functions.
- Set x = y and x = y^2 equal to each other: y = y^2.
- Rearrange the equation: y^2 - y = 0.
- Factor out y: y(y - 1) = 0.
- Solution 1: y = 0.
- Solution 2: y - 1 = 0, which means y = 1.
- So the limits of integration for y are 0 and 1.
- The limits of integration for x are x = y and x = y^2, since the region R is bounded by y=x and y=x^2.
2. Set up the double integral.
- The volume of the solid is given by the double integral of f(x,y) over the region R.
- V = ∬ f(x,y) dA, where dA represents the infinitesimal area in the xy-plane.
- In this case, f(x,y) = 2x^2y.
3. Set up the limits of integration.
- The limits of integration for y are 0 and 1, as determined earlier.
- The limits of integration for x are x = y and x = y^2, as determined earlier.
4. Evaluate the double integral.
- Integrate f(x,y) = 2x^2y with respect to x first, then with respect to y.
- ∫[y, y^2] ∫[0, 1] 2x^2y dx dy
- ∫[y, y^2] (2/3)x^3y |[0, 1] dy
- ∫[y, y^2] (2/3)(y^3)y - (2/3)(0^3)y dy
- ∫[y, y^2] (2/3)y^4 dy
- (2/3)(1/5)y^5 |[y, y^2]
- (2/3)(1/5)y^5 - (2/3)(1/5)y^5
- (2/3)(1/5)(y^5 - y^5)
- (2/15)(y^5 - y^5)
- (2/15)(0)
- 0
5. The volume of the solid is 0.
- This means that the solid is empty or degenerate.
Note: If you made a mistake in the calculation or if the result doesn't seem correct, double-check your work and ensure that the function and region are defined correctly.