help with calculus problem??

Question

find the area of defined regions

The vertical lines x=0 and x=1, the x-axis and the graph of y=x2

 

 

 

The vertical lines x=0 and x=1, the x-axis and the graph of y=2x2

Answer 1

For the first part, draw out each portion on the graph and you will notice that we are just integrating ƒ(x)=x² on the interval [0,1]:

∫₀¹x²dx = (x³/3)|₀¹ = 1/3-0/3 = 1/3

Part 2 is analogous. Our bounds are the same but we are integrating ƒ(x)=2x² now. But notice that we can pull the constant term 2 out of the integral:

2 ∫₀¹x²dx = 2(x³/3)|₀¹ = 2(1/3-0/3) = 2/3