The area under a curve.

Step 1 of 5

The area under a line

The second question of calculus is how much area lies between a curve and the $x$⁠-axis. Start with a line, where you already know how: the board shows $y = x$⁠, and under it, from $x = 0$⁠ to $x = 4$⁠, sits a triangle. Answer, then choose The area under Look inside.

Step 2 of 5

Narrow rectangles

Under a curve there is no triangle. Instead, cut the area into narrow rectangles, each as tall as the curve at its left edge, and add up their areas. That is a Riemann sum.

The board shows $y = x^2$⁠, from $0$⁠ to $3$⁠, in three rectangles $1$⁠ wide.

Step 3 of 5

More rectangles

The rectangles in the last step left pieces of area out. What happens if you cut the same stretch into $30$⁠ narrow rectangles?

Step 4 of 5

The antiderivative

The narrower the rectangles, the closer their sum comes to one number: the exact area, the integral, written $\displaystyle\int_0^3 x^2\,dx$⁠. You can get that number without rectangles, from an antiderivative: a function whose derivative is the curve. The derivative of $\dfrac{x^3}{3}$⁠ is $x^2$⁠, so

$\int_0^3 x^2\,dx = \frac{3^3}{3} - \frac{0^3}{3}$$\int_0^3 x^2\,dx =$$\frac{3^3}{3} - \frac{0^3}{3}$⁠

In general: raise the power by one and divide by the new power. It is the reverse of the rule for derivatives from the last Lesson.

Step 5 of 5

Below the axis

The board shows $y = x - 2$⁠, from $0$⁠ to $4$⁠. Part of the area is above the $x$⁠-axis and part below it. The integral counts area below the axis as negative: there the height $y$⁠ is negative, and width times a negative height is a negative number. On the board the negative area is pink.