How Do You Solve for a and b in an Integral Equation?

[ProBasket]

$$\int_1^{13} f(x) - \int_1^{10} f(x) = \int_a^{b} f(x)$$

Question:

where a = ____ and b = ____

what is it asking me extactly? is it just telling me to substract the b's and a's? if so, a = 0 and b = 3, but it's wrong.

 

[BobG]

Draw f(x) on a graph (it doesn't really matter what f(x) is, just draw a curvy line from 0 to 13). Draw a vertical line from x=1 to the curve. Draw a vertical line from 10 to the curve. Draw a vertical line from 13 to the curve.

Your first integral is the area under the curve from 1 to 13. The second integral subtracts the area under the curve from 1 to 10 (shade it in). What are you left with. This is the area you want to calculate.

You're right you have an interval of 3, but unless f(x) is a straight horizontal line, the area from 10 to 13 won't necessarily be the same as the area from 0 to 3.

 

[wais]

The question is asking you to find the values of a and b that would make the equation true. Since the integral on the left side of the equation has a range of 1 to 13, and the integral on the right side has a range of a to b, we can set the two equal to each other and solve for a and b. In this case, a = 1 and b = 13. This means that the range of the integral on the right side of the equation should also be from 1 to 13 in order for the equation to be true. Simply subtracting the b's and a's is not enough to solve this problem.