Calculate The Integral by Fundamental Theorem of Calculus

What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus consists of two parts that establish a deep connection between differentiation and integration:

1. First Part: If a function f is continuous on the closed interval [a, b] and F is an antiderivative of f on [a, b], then the definite integral of f from a to b is equal to F(b) - F(a).
2. Second Part: If f is continuous on an open interval containing a and b, then the function g defined by g(x) = ∫ from a to x of f(t) dt has a derivative at every point of the interval, and g'(x) = f(x).

This theorem allows us to evaluate definite integrals by finding antiderivatives, which is often much easier than using the limit definition of integration.

How to Use the Fundamental Theorem of Calculus

To calculate a definite integral using the Fundamental Theorem of Calculus, follow these steps:

1. Find an antiderivative F(x) of the integrand f(x).
2. Evaluate F at the upper limit (b) and the lower limit (a).
3. Subtract F(a) from F(b) to get the definite integral value.

For example, to integrate x² from 0 to 1:

1. Find an antiderivative: ∫x² dx = (1/3)x³ + C
2. Evaluate at 1: (1/3)(1)³ = 1/3
3. Evaluate at 0: (1/3)(0)³ = 0
4. Subtract: 1/3 - 0 = 1/3

Example Calculation

Let's calculate ∫₁² 3x² dx using the Fundamental Theorem of Calculus.

1. Find an antiderivative: ∫3x² dx = x³ + C
2. Evaluate at 2: (2)³ = 8
3. Evaluate at 1: (1)³ = 1
4. Subtract: 8 - 1 = 7

The value of the integral is 7.

Common Mistakes to Avoid

When using the Fundamental Theorem of Calculus, be careful about these common errors:

• Forgetting to evaluate the antiderivative at both limits (a and b).
• Using the wrong antiderivative (one that doesn't match the integrand).
• Ignoring the constant of integration (it cancels out, but the antiderivative must be correct).
• Assuming the integrand is continuous where it shouldn't be (the theorem requires continuity).

Applications in Real Life

The Fundamental Theorem of Calculus has many practical applications:

• Calculating areas under curves in physics and engineering.
• Determining distances traveled by objects with varying speeds.
• Finding total work done by variable forces.
• Calculating average values of functions.
• Solving problems in economics and finance involving rates of change.

For example, in physics, the Fundamental Theorem of Calculus allows us to find the displacement of an object by integrating its velocity function.