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Use The Fundamental Theorem to Calculate The Following Integral Exactly:

Reviewed by Calculator Editorial Team

The Fundamental Theorem of Calculus connects differentiation and integration, allowing us to calculate definite integrals exactly when we know an antiderivative. This powerful theorem simplifies many complex calculations in mathematics, physics, and engineering.

What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus consists of two related 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 ∫ from a to b of f(x) dx = F(b) - F(a).
  2. Second Part: If f is continuous on an open interval containing a, and F(x) = ∫ from a to x of f(t) dt, then F is differentiable on that interval and F'(x) = f(x).

This theorem allows us to evaluate definite integrals by finding antiderivatives, which is often much easier than computing the integral directly.

How to Use the Theorem to Calculate Integrals

Step-by-Step Process

  1. Identify the integrand - The function you're integrating (f(x)).
  2. Find an antiderivative - Find a function F(x) such that F'(x) = f(x).
  3. Apply the theorem - Calculate F(b) - F(a) to find the exact value of the integral.
  4. Verify the result - Check that F'(x) equals the original integrand.

Formula: ∫ from a to b of f(x) dx = F(b) - F(a) where F'(x) = f(x)

Key Considerations

  • The integrand must be continuous on the closed interval [a, b].
  • The antiderivative must be found correctly to ensure accuracy.
  • For definite integrals, the limits of integration must be constants.

Example Calculation

Let's calculate ∫ from 0 to 2 of 3x² dx exactly using the Fundamental Theorem.

  1. Identify the integrand: f(x) = 3x²
  2. Find an antiderivative: F(x) = x³ (since d/dx of x³ is 3x²)
  3. Apply the theorem: F(2) - F(0) = (2)³ - (0)³ = 8 - 0 = 8
  4. Verification: The derivative of x³ is indeed 3x², which matches our integrand.

Result

The exact value of ∫ from 0 to 2 of 3x² dx is 8.

Common Mistakes to Avoid

  • Incorrect antiderivative: Always double-check that the derivative of your antiderivative matches the integrand.
  • Forgetting limits: Remember to evaluate the antiderivative at both the upper and lower limits.
  • Sign errors: Be careful with negative limits of integration.
  • Discontinuous functions: The theorem requires the integrand to be continuous on the closed interval.

Tip: When in doubt, verify your antiderivative by differentiating it to ensure it matches the original function.

FAQ

Can the Fundamental Theorem be used for all integrals?

No, the theorem requires that the integrand is continuous on the closed interval and that you can find an antiderivative. For integrals that don't meet these conditions, other methods may be needed.

What if I can't find an antiderivative?

If you can't find an antiderivative, you may need to use numerical methods or approximation techniques to estimate the integral's value.

How does this theorem relate to the Mean Value Theorem?

The Mean Value Theorem for Integrals states that for a continuous function f on [a, b], there exists a c in [a, b] such that f(c) = (1/(b-a))∫ from a to b of f(x) dx. This is a consequence of the Fundamental Theorem.