Area Between Two Curves - Calculus Step by Step

Find the area between f(x) = x + 2 and g(x) = x² from x = -1 to x = 2 by integrating the difference (top - bottom): area = F(2) - F(-1) = 9/2.

area-between-curves The area between curves f (top) and g (bottom) on [a, b] equals ∫_a^b [f(x) - g(x)] dx = F(b) - F(a), where F is an antiderivative of the integrand f - g. The endpoints a, b are where the curves intersect.

Example

Integrate top minus bottom to compute the exact area between curves.

highlighted = computed this step

Step 1 — Set up

Set up the interval from -1 to 2.

[ -1 , 2 ]\quad f(x)= x+2 \quad g(x)= x^{2}

Step 2 — Identify top and bottom

Identify x plus 2 as top and x squared as bottom.

\text{top}= \hlmath{x+2} \quad \text{bottom}= \hlmath{x^{2}}

Step 3 — Subtract curves

Subtract top minus bottom for the integrand.

\text{top}-\text{bottom}= \hlmath{x+2-x^{2}}

Step 4 — Antiderivative

Find an antiderivative of the difference.

F(x)= \hlmath{\frac{x^{2}}{2}+2x-\frac{x^{3}}{3}}

Step 5 — Evaluate upper bound

Evaluate F at 2 to get 10 over 3.

F( 2 )= \hlmath{\frac{10}{3}}

Step 6 — Evaluate lower bound

Evaluate F at -1 to get -7 over 6.

F( -1 )= \hlmath{\frac{-7}{6}}

Step 7 — Area

Subtract 10 over 3 minus -7 over 6 to get 9 over 2.

\frac{10}{3} - \frac{-7}{6} = \hlmath{\frac{9}{2}}