How to find the area under a curve?

Approximating with rectangles to exact integration

To determine the total area under a curve, picture slicing the region beneath the graph into vertical strips. You can approximate each strip as a rectangle whose height equals the value of the function at that point. Adding up the areas of these rectangles gives an estimate known as a Riemann sum.

As you increase the number of rectangles, their widths shrink toward zero. The gap between the flat tops of the rectangles and the smooth curve disappears. Taking the limit as the number of rectangles approaches infinity turns this sum into a definite integral, giving the exact geometric area.

The three-step method to calculate the area

First, determine the lower limit a and upper limit b. If the problem specifies boundaries like x = 1 and x = 4, use those numbers. If it asks for the area enclosed by the curve and the x-axis, set f(x) = 0 and solve for x to identify the x-intercepts.

Second, find an antiderivative F(x) of your function f(x). You do not need to include the constant of integration C because it cancels out when you subtract.

Third, apply the Fundamental Theorem of Calculus: compute F(b) - F(a). For example, to find the area under f(x) = 3x^2 from x = 1 to x = 3, take the antiderivative F(x) = x^3. Then evaluate F(3) - F(1) = 3^3 - 1^3 = 27 - 1 = 26 square units.

What to watch for: signs and boundaries

This direct method assumes f(x) remains on or above the x-axis across the entire interval from a to b. If the function drops below the x-axis, the definite integral assigns that lower section a negative value. To compute total geometric area, you must split the integral at every x-intercept and take the absolute value of any region lying below the axis.