Can someone look at my work and see what i did wrong. I did this 100 times but i keep getting it wrong
Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region.
2y=3x^(1/2) , y=4 and 2y+1x=4
y=3(x^(1/2))/2, y=4 , y=2-(1/2)x
Intersection points = -4,1,64/9
Integral from -4 to 1
4x-2x-(x^2)/4
Integral from 1 to 64/9
4x-3(x^(5/2))/5
please help i've been working on this for hours
go to wolframalpha . com and enter
graph x=4y^2/9,x+2y=4,y=4,y=3/2,x=1
to see the curves involved.
The area of interest is a little v-shaped chunk from
(-4,4) down to (1,3/2) and back up to (64/9,4)
If you integrate on x, then the integrands use the limits you have, but
the height in each case is 4-y, for the appropriate f(x)
4 - (4-x)/2 from -4 to 1 = 25/4
4 - 3/2 sqrt(x) from 1 to 64/9 = 175/27
match anything you got?
Thank you soooo much steve i finally got it right!!!! :D
To find the region enclosed by the given curves and calculate its area, you need to follow these steps:
1. Plot the given curves on a graph to visualize the region. In this case, you have three equations: 2y = 3x^(1/2), y = 4, and 2y + x = 4.
2. To decide whether to integrate with respect to x or y, examine the curves and determine if the region is better suited for horizontal or vertical integration. Looking at the given equations, it seems that the curves intersect horizontally. Therefore, it is more appropriate to integrate with respect to x.
3. Find the intersection points of the curves by setting each pair of equations equal to each other. In this case, you need to find the points where 2y = 3x^(1/2) and 2y + x = 4 intersect. Solve these equations simultaneously to find the x-values that correspond to the intersection points. The values you listed, -4, 1, and 64/9, are the correct intersection points.
4. Determine the limits of integration for x. The limits of integration are the x-values where the curves intersect, which are -4 and 1 in this case.
5. Set up the integral for the area calculation. Since you decided to integrate with respect to x, you should integrate the difference between the upper and lower curves with respect to x over the limits -4 to 1.
6. Evaluate the integral. Plug in the expressions for the upper and lower curves and integrate with respect to x over the given limits.
From your work, it seems that you correctly set up the integral and found the intersection points. However, there seems to be a slight error in the integration itself.
For the first integral from -4 to 1, it should be:
∫ (4 - 2x - (x^2)/4) dx
For the second integral from 1 to 64/9, it should be:
∫ (4x - 3(x^(5/2))/5) dx
Make sure to double-check your integration steps and calculations to ensure accuracy.