0
$\begingroup$

I am asked to find the region bounded by the following curves and I keep getting a different answer than the solutions packet. These are the curves... x = -2 ; x = 3 ; f(x) = -x^2 + 4 ; y = 0

The solutions packet says the answer is 13. Meanwhile my answer is 25/3. I really just want to know how the answer packet gets to 13....

1 Answers 1

1

You did,

$$\int_{-2}^{3} (-x^2+4) dx$$

But because the integral gives $\text{signed}$ area what we really want is,

$$\int_{-2}^{3} |-x^2+4| dx$$

Now note that $-x^2+4 >0$ means $4>x^2$ or $2>|x|$. This means the integrand without absolute value is already positive for $-2

$$\int_{-2}^{2} (-x^2+4) dx+\int_{2}^{3} -(-x^2+4) dx$$

Sure enough this gives $13$.

  • 1
    Wow!! I would never have thought of that on my own! Thank you so much!! – 2017-02-17
  • 0
    No problem. I'm sure with some time you would have realized that some of the area below is below the $x$ axis and counted as negative by your integral, and adjusted accordingly. – 2017-02-17