Find the area of the region bounded by the parabola y = 2x2, the tangent line to this parabola at (2, 8), and the x-axis.

Question

I am having trouble overall with the topic (Find the area of shaded region) in Calc 2. I would like to have a clear understanding of how to find the answer to this problem. A step by step solution would really be helpful. Thank You!

Answer 1

Hi Leslie,

              I'll try and get you started but it is difficult to do without a graphic tool.

The upper boundary is the parabola y=2x² 

Part of the lower boundary is the tangent line to the parabola at (2,8) so our first task is to find the equation for this line and then find out where this line intersects the x axis.

 

The slope of this line is dy/dx @x=2, dy/dx =4x and at x=2 is equal to 8. So the tangent line is (y-8)=8(x-2) and rearranging gives y=8x-8. This line intersect the x axis at x=1 which is one of the limits of our integral.

 

The area is A=∫₀¹2x²dx +∫₁²(2x²-8x+8)dx I'll leave the integration to you.

Hope this helps

 

Jim