How to proof calculus that area under a curve is anti-differentiation of the function
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The Fundamental Theorem of Calculus connects differentiation and integration. We want to prove that if F of x equals the integral from a to x of f of t dt, then F prime of x equals f of x. In other words, the area function is the antiderivative of the original function.
Step one: Define the area function. Let f be a continuous function on the interval from a to b. We define F of x as the area under the curve f of t from the fixed point a to the variable point x. This gives us F of x equals the integral from a to x of f of t dt. Our goal is to prove that F prime of x equals f of x.
Step two: Compute the derivative of F. The derivative of F of x is defined as the limit as h approaches zero of F of x plus h minus F of x, all divided by h. When we substitute our definition of F, we get the limit of one over h times the integral from a to x plus h minus the integral from a to x. Using properties of definite integrals, this simplifies to the limit of one over h times the integral from x to x plus h of f of t dt.
Step three: Apply the Mean Value Theorem for integrals. Since f is continuous on the interval from x to x plus h, the Mean Value Theorem guarantees that there exists a point c sub h between x and x plus h such that the integral from x to x plus h of f of t dt equals f of c sub h times h. This means the area under the curve equals the area of a rectangle with height f of c sub h and width h. Therefore, F prime of x becomes the limit as h approaches zero of f of c sub h.
Step four: Evaluate the limit and reach our conclusion. As h approaches zero, the interval from x to x plus h shrinks to the single point x. Since c sub h is trapped between x and x plus h, we have c sub h approaching x. Because f is continuous, the limit of f of c sub h as h approaches zero equals f of x. Therefore, F prime of x equals f of x. We have proven that the area function F of x is indeed the antiderivative of f of x. This completes the proof of the Fundamental Theorem of Calculus Part One.