Areas under Curves
Finding Areas under Curves
- To find the area under a curve, get the two limits and integrate once for each of your limits.
- Then subtract one from the other.
- If the area is over the x axis, the area should be positive
- otherwise it should be negative.
Integration under curves example 1
Area = \(\int ^2_{-1}x^2 \, dx\)
= \(\left[ \frac{1}{2}x^3 \right]^2_{-1}\)
=\(\frac{1}{3}(2)^3-\left( \frac{1}{3}(-1)^3 \right)=\frac{9}{3}=3\)
Integration under curves example 2
Area = \(\int ^0_{-4}(4x-3x^2-x^3) \, dx\)
= \(\left[ 2x^2-x^3-\frac{x^4}{4} \right]^0_{-4}\)
= \((0)-\left( 2(-4)^2-(-4)^3-\frac{(-4)^4}{4} \right)\)
= \(-32+64-64\)
=\(-32\)