Mean Value And Volume Of Revolution

Question 1

What is the mean value n

Answer 1

Basically you want to make a RECTANGLE, such that the width of the integral is the SAME, the area is the SAME

Thus if integral = width x height, and width = b-a (limits)
Mean value (height) = 1/b-a x integral!

Question 2

What happens if it’s an ODD FUNCTION with the same limits on either side of origin? Why is the mean value 0

Answer 2

• as odd function
• the only way for the value of the area of integral = to the area of height is if height is 0

This because area of integral will be 0 (negative + positive)
And so height must be 0

Question 3

How to do further polynomial division

Answer 3

When you have like x and y and numbers seem to work and it will just make algebra hella complex if expand, try divide

When you make a term that’s not in original, just add it in the original and SUBTRACT IT STRAIGHT AWAY. This is allowed as you did nothing!

Then see if dividing now kicks each term out as that’s all you need to do

Always multiply to double check!

Test 3b-a divide 9b 2 - a 2 +15b -5a!

Question 4

Don’t lack, if it says rotate around y axis, but they gave x values? What To do?

Answer 4

Still sub in x to get y limits

Now still integrate x 2 wrt to y, don’t fall for trick

Question 5

Demmeebr fir General extra integration

Answer 5

If it’s in any power of 1/2 and looks like standsrd sub thst one in

Question 6

Rememebr tan intehral can be helpful

Answer 6

1/a2 -x2

Can tell because tan is just a +x, so make it minus and tanh!