Finding the Area Between Two Functions on a Given Interval

[Niaboc67]

Homework Statement

Find the area of the region between the graphs of: y=x^3+3x^2+5x and y = x^3+2x^2+7x on the interval [-1,2]

The Attempt at a Solution

I am not entirely sure what they mean by the REGION between the graphs, is this the region which encloses an area when the two functions cross over each other? not entirely sure what that means. I digress.

To find where they overlap I set them equal to each other and then isolated one entire side to zero. and got: x^2 - 2x = 0
So is this like my "main function" that I need in order to find the area. Is this the function that represents the area in between the aforementioned functions? If so, then I'd have to get the area and to get the area of any function I am guessing I'll have to take it's integral and since it's been defined on the interval [-1,2] I use those two points as a and b for my integral.

Therefore,
∫[from -1 to 2] |x^2 - 2x|dx

I know area is always positive so is it a good idea to always use absolute values when taking a functions integral? And is the dx always needed all the other problems I did seem to use that dx symbol.From that point I get a bit stuck as to what to do next

.
Please help
Thank you

 

[andrewkirk]

The area between the two curves is positive regardless of which curve is uppermost at any value of x. Hence it is equal to the integral between the two x limits of the absolute value of the difference between the two functions.

In practice the easiest way to evaluate it is often to break up the integral into parts that run between consecutive crossover points. Evaluate the integral of the difference of the functions over each of those subintervals (don't put absolute value signs inside the integrals), then add up the absolute values of the results.

 

[Ray Vickson]

Homework Statement

Find the area of the region between the graphs of: y=x^3+3x^2+5x and y = x^2+2x^2+7 on the interval [-1,2]

The Attempt at a Solution

I am not entirely sure what they mean by the REGION between the graphs, is this the region which encloses an area when the two functions cross over each other? not entirely sure what that means. I digress.

To find where they overlap I set them equal to each other and then isolated one entire side to zero. and got: x^2 - 2x = 0
So is this like my "main function" that I need in order to find the area. Is this the function that represents the area in between the aforementioned functions? If so, then I'd have to get the area and to get the area of any function I am guessing I'll have to take it's integral and since it's been defined on the interval [-1,2] I use those two points as a and b for my integral.

Therefore,
∫[from -1 to 2] |x^2 - 2x|dx

I know area is always positive so is it a good idea to always use absolute values when taking a functions integral? And is the dx always needed all the other problems I did seem to use that dx symbol.From that point I get a bit stuck as to what to do next

.
Please help
Thank you

Is there a typo in your second curve y = x^2+2x^2+7? Should that be y = x^3 + 2 x^2 + 7, or should it be y = x^2 + 2x + 7?

 

[Niaboc67]

@Ray Vickson Has been corrected.