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Area Between Curves

Find where the curves meet, then add up thin strips of height top minus bottom — splitting wherever the curves swap places.

Step 1 of 24

— f(x) = x + 2— g(x) = x²■ f on top■ g on top

Speed

The line y = x + 2 and the parabola y = x² cross twice and close off a region. What is its area? First we need to know where it starts and ends.

f(x) = x + 2, \qquad g(x) = x^2, \qquad \text{crossings at } x = -1,\ 2

\text{Area} = \int_{-2}^{-1} (x^2 - x + 2)\,dx + \int_{-1}^{2} (x + 2 - x^2)\,dx + \int_{2}^{3} (x^2 - x + 2)\,dx = 1.8333 + 4.5 + 1.8333 = 8.1667

\int_{-2}^{3} (f - g)\,dx = 0.8333 \quad \text{(crossings make pieces cancel)}

Find where the curves meet, then add up thin strips of height top minus bottom — splitting wherever the curves swap places.

Step 1 of 24

— f(x) = x + 2— g(x) = x²■ f on top■ g on top

Speed

The line y = x + 2 and the parabola y = x² cross twice and close off a region. What is its area? First we need to know where it starts and ends.

f(x) = x + 2, \qquad g(x) = x^2, \qquad \text{crossings at } x = -1,\ 2

\text{Area} = \int_{-2}^{-1} (x^2 - x + 2)\,dx + \int_{-1}^{2} (x + 2 - x^2)\,dx + \int_{2}^{3} (x^2 - x + 2)\,dx = 1.8333 + 4.5 + 1.8333 = 8.1667

\int_{-2}^{3} (f - g)\,dx = 0.8333 \quad \text{(crossings make pieces cancel)}