Question

Find the range of the function F(x)= integral from [-4, x] sqrt(16-t^2) dt

Answer 1

hint: y = sqrt(16-x^2) is a semi-circle with radius 4

Answer 2

so [-4,4]?

Answer 3

the integral represents the area under the curve, which in this case is the area of a part of the semi-circle or all of it

Answer 4

what is the smallest area possible?

Answer 5

0

Answer 6

yes, so that is the lower bound of the range

Answer 7

what is the largest area possible? ie, what is the area of the semi-circle?

Answer 8

[0, 4] because its asking for range and that encompasses the highest point

Answer 9

right?

Answer 10

4pi

Answer 11

area of semi-circle formula

\[\Large A = \frac{\pi*r^2}{2}\]

Answer 12

in this case, r = 4

Answer 13

so the largest area is 8pi

Answer 14

yes, which is the upper bound of the range

Answer 15

therefore, the range is from 0 to 8pi

Answer 16

ahhh ok

Answer 17

in interval notation, you'd write \(\Large [0, 8\pi]\)

Answer 18

I understand. I'm posting a few more questions. Mind helping out?

Answer 19

Sure, go ahead