# Let f(x)= 0, if, x < 0, x, if, 0 \leq x \leq 10, 20-x ,if, 10 < x \leq 20, 0, if, x > 20; and ...

## Question:

Let {eq}f(x)= 0, if, x < 0, x, if, 0 \leq x \leq 10, 20-x ,if, 10 < x \leq 20, 0, if, x > 20; {/eq}

and {eq}g(x)=\int_0^xf(t)dt. {/eq}

Find an expression for {eq}g(x) when 10 < x < 20 . {/eq}

1. {eq}g(x)=20x-\frac {1}{2}x^2-50 {/eq}

2. {eq}g(x)=20x-\frac {1}{2}x^2-100 {/eq}

3. {eq}g(x)=20x-\frac {1}{2}x^2 {/eq}

## Integration:

Integration is the process of finding the function f(x) given the function f'(x). For finding the function g(x) when x is between 10 and 20 we have to integrate f(x) over the range 0 to x for that value of function f(x) for which x is between 10 and 20.

{eq}g(x)=\int_0^xf(t)dt\\ g(x) = \int_0^x (20 -t) dt \\ g(x) = (20t - \frac{t^2}{2})_0^x\\ g(x) = 20x - \frac{x^2}{2} {/eq}

Thus option (3) is correct.