The definite integral is a fundamental concept in calculus that represents the signed area under a curve. Given a continuous function f of x equals x squared plus 1 on the interval from 0 to 3, we want to find the area of this shaded region. But how do we calculate this area when it's not a simple geometric shape like a rectangle or triangle? The definite integral, written as the integral from a to b of f of x dx, gives us the exact answer to this problem.
To approximate the area under the curve, we can use rectangles. This method is called a Riemann sum. We divide the interval from a to b into n equal parts, each with width delta x equals b minus a over n. For each subinterval, we create a rectangle whose height is determined by the function value at a chosen point. The sum of all rectangle areas gives us an approximation. Let's start with 4 rectangles and see how the approximation improves as we increase the number of rectangles.
The key insight is the limiting process. As we increase the number of rectangles n towards infinity, the width of each rectangle approaches zero, and our approximation becomes increasingly accurate. The definite integral is defined as this limit: the integral from a to b of f of x dx equals the limit as n approaches infinity of the Riemann sum. Watch as we progress from 4 rectangles to 8, then 16, 32, and finally 64 rectangles. Notice how the approximation converges to the exact area under the curve.
The geometric interpretation of definite integrals involves signed areas. When the function is above the x-axis, the integral gives positive area. When the function is below the x-axis, the integral gives negative area. For functions that cross the x-axis, the definite integral represents the net area - the sum of positive and negative contributions. This signed area concept is fundamental to understanding how definite integrals work in various applications.
Let's work through a complete example. We want to calculate the definite integral from 0 to 2 of 3x squared plus 2x plus 1 dx. First, we find the antiderivative: F of x equals x cubed plus x squared plus x. Then we apply the Fundamental Theorem of Calculus: F of 2 minus F of 0 equals 8 plus 4 plus 2 minus 0, which equals 14. This gives us the exact area under the curve, confirming our understanding of definite integrals.