What is The area of the region bounded between y equals x2 plus 2x-3 and y equals 4x plus 45?

Answer 1

First, you'll need to find the x co-ordinates of the two points where that line and curve intersect:

y = x2 + 2x - 3
y = 4x + 45

∴ x2 + 2x - 3 = 4x + 45
∴ x2 - 2x - 48 = 0
∴ (x - 8)(x + 6) = 0
∴ x1 = -6, x2 = 8

Now let's figure out which one of the two is higher on the y axis. This can be done by taking any point within the range and seeing which of the two equations has a higher value. Zero is between those points, and a fairly easy one to calculate, so let's try that:

02 + 2·0 - 3 = -3
4·0 + 45 = 45

So the line has the higher value. This means we'll need to take the area under the curve, and subtract it from the area under the line. This can be done by taking their definite integrals for the range -6 to 8, and subtracting the result for the curve from the result for the line.

A = ∫-68 (4x + 25) dx - ∫-68 (x2 - 2x - 48) dx
∴ A = (2x2 + 25x)|-68 - (x3/3 - x2 - 48x)|-68
∴ A = [(2·64 + 25·8) - (2·36 + 25·-6)] - [(512/3 - 64 - 48·8) - (-216/3 - 36 - 48·-6)]
∴ A = (128 + 200 - 72 - 150) - (170 2/3 - 64 - 384 +72 + 36 - 288)
∴ A = 106 - (-457 1/3)
∴ A = 563 1/3