Another problem involving computing area with integrals

In summary, the task is to sketch the region enclosed by the curves x+y^2=6 and x+y=0, and to determine whether to integrate with respect to x or y to find the area. It is found that integrating with respect to y is easier, resulting in an area of 49/3. However, if integrating with respect to x is preferred, the formula for the area of the typical element will need to be adjusted due to the varying y values at the bottom of the area elements.
  • #1
haydn
27
0

Homework Statement



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.

x+y^2=6

x+y=0

Homework Equations



none, just using integrals...

The Attempt at a Solution



I sketch the two curves and find they intersect at x=-3 and x=2. Looking at it I want to integrate with respect to x.

y=[tex]\sqrt{-x+6}[/tex] appears to be the top function, while y=-x is the lower function.

I set up an integral from -3 to 2 that consists of: [tex]\sqrt{-x+6}[/tex] - (-x) dx.

I evaluate using substitution and keep getting 10.16, which the homework website I'm using says is wrong. Not sure what I'm doing incorrectly here.
 
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  • #2
It's a lot easier if you integrate w.r.t y (i.e., use horizontal area elements rather than vertical elements). If you do it this way there are no radicals to deal with. The area I get is 49/3.

You can find the area using vertical area elements, but the y values at the bottoms of your area elements change. For one interval, the function at the bottom is y = -x; for the other interval, the function at the bottom is y = -sqrt(-x + 6). This means that your formula for the area of the typical element has to change, and that means you need to integrals.
 
  • #3
BTW, this problem is very similar to the one posted yesterday by vipertongn.
 
  • #4
Hey thanks a lot. I get it now. I got the correct answer.
 

Related to Another problem involving computing area with integrals

1. What is the formula for computing area with integrals?

The formula for computing area with integrals is ∫f(x) dx, where f(x) is the function representing the height of the area and dx is the infinitesimal width of each rectangle used to approximate the area.

2. How do you determine the limits of integration for computing area with integrals?

The limits of integration are determined by the boundaries of the region being measured. The lower limit is typically the smallest x-value of the region, and the upper limit is the largest x-value.

3. Can integrals be used to compute the area of irregular shapes?

Yes, integrals can be used to compute the area of irregular shapes by breaking the shape into smaller, simpler shapes and using multiple integrals to compute the area of each individual shape.

4. What is the difference between definite and indefinite integrals when computing area?

A definite integral is used to find the exact area under a curve between two specific limits, while an indefinite integral is a general function that represents the antiderivative of another function, and does not have specific limits.

5. How can integrals be used for more than just computing area?

Integrals can also be used to compute the volume of 3D shapes, determine the average value of a function, and solve optimization problems in various fields such as physics, economics, and engineering.

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