To find the area of a circle using calculus, we divide the circle into infinitesimally thin concentric rings. Each ring has a radius r and a small thickness dr. We can think of each ring as a very thin rectangle when unrolled, with length equal to the circumference 2πr and width equal to dr. The area of this thin ring is approximately 2πr times dr. To find the total area, we integrate this expression from r equals 0 to r equals capital R, which is the radius of the circle.
Now let's set up the integral to find the area of the circle. For each thin ring with radius r and thickness dr, the area is approximately 2πr times dr. To find the total area, we integrate this expression from r equals 0 at the center to r equals capital R at the edge. This gives us the integral from 0 to R of 2πr dr. We can visualize this as the area under the curve 2πr from 0 to R. This is a simple integral to evaluate since we're integrating r with respect to r.
Now let's evaluate the integral to find the area of the circle. We have the integral from 0 to R of 2πr dr. First, we can pull the constant 2π outside the integral. Then we integrate r with respect to r, which gives us r squared over 2. We evaluate this from 0 to R, which gives us R squared over 2 minus 0 squared over 2. The second term is zero, so we're left with 2π times R squared over 2, which simplifies to π times R squared. This is the well-known formula for the area of a circle with radius R.
Let's see how we can approximate the area of a circle by dividing it into a finite number of rings. For each ring, we can calculate its area as 2π times its average radius r_i times its thickness Δr. The total area is approximately the sum of all these ring areas. With just 4 rings, we get a rough approximation. With 8 rings, the approximation gets better. With 16 rings, it's even more accurate. As we increase the number of rings to infinity and make each ring infinitesimally thin, the sum becomes the integral from 0 to R of 2πr dr, which equals πR². This is how calculus gives us the exact area of a circle.
To summarize what we've learned: The area of a circle can be calculated using calculus by dividing it into infinitesimal rings. Each ring has an area equal to its circumference times its thickness, which is 2πr times dr. By integrating this expression from the center to the edge of the circle, we get the formula πR². This approach demonstrates how calculus can be used to find areas of curved shapes by breaking them down into infinitesimal pieces and summing them up through integration. As the number of rings approaches infinity and their thickness approaches zero, the approximation becomes exact.