Question 1

What is the Fundamental Theorem of Calculus?

Accepted Answer

The Fundamental Theorem of Calculus (FTC) is a central theorem that connects differentiation and integration. It has two parts: Part 1 states that if F(x) = ∫f(t)dt from a to x, then F'(x) = f(x). Part 2 states that ∫f(x)dx from a to b = F(b) - F(a), where F is any antiderivative of f.

Question 2

How do you apply the Fundamental Theorem of Calculus Part 1?

Accepted Answer

FTC Part 1 states that d/dx[∫f(t)dt from a to x] = f(x). This means if you have an integral with a variable upper limit, the derivative of that integral is the original function. For example, d/dx[∫t²dt from 0 to x] = x². This connects integration and differentiation as inverse operations.

Question 3

How do you apply the Fundamental Theorem of Calculus Part 2?

Accepted Answer

FTC Part 2 states that ∫f(x)dx from a to b = F(b) - F(a), where F is any antiderivative of f. To use this: 1) Find an antiderivative F(x) of f(x), 2) Evaluate F(b) and F(a), 3) Calculate F(b) - F(a). For example, ∫x²dx from 0 to 2 = [x³/3]₀² = 8/3 - 0 = 8/3.

Question 4

What is the difference between definite and indefinite integrals?

Accepted Answer

A definite integral ∫f(x)dx from a to b gives a specific numerical value (the area under the curve). An indefinite integral ∫f(x)dx gives a function (the antiderivative) plus a constant C. The FTC Part 2 connects them: the definite integral equals the antiderivative evaluated at the bounds.

Question 5

How do you find antiderivatives using the power rule?

Accepted Answer

The power rule for integration states that ∫xⁿdx = xⁿ⁺¹/(n+1) + C, where n ≠ -1. For example: ∫x²dx = x³/3 + C, ∫x³dx = x⁴/4 + C, ∫x⁰dx = ∫1dx = x + C. This is the reverse of the power rule for differentiation: d/dx[xⁿ] = nxⁿ⁻¹.

Question 6

What are common integration techniques?

Accepted Answer

Common integration techniques include: 1) Power rule: ∫xⁿdx = xⁿ⁺¹/(n+1) + C, 2) Constant multiple rule: ∫cf(x)dx = c∫f(x)dx, 3) Sum rule: ∫[f(x) + g(x)]dx = ∫f(x)dx + ∫g(x)dx, 4) Substitution: ∫f(g(x))g'(x)dx = F(g(x)) + C, 5) Integration by parts: ∫udv = uv - ∫vdu.

Question 7

How do you evaluate definite integrals using FTC?

Accepted Answer

To evaluate ∫f(x)dx from a to b using FTC: 1) Find any antiderivative F(x) of f(x), 2) Apply FTC Part 2: ∫f(x)dx from a to b = F(b) - F(a), 3) Calculate F(b) and F(a), 4) Find the difference F(b) - F(a). For example, ∫x²dx from 1 to 3 = [x³/3]₁³ = 27/3 - 1/3 = 26/3.

Question 8

What are the applications of the Fundamental Theorem of Calculus?

Accepted Answer

FTC applications include: 1) Calculating areas under curves, 2) Finding net change in quantities over time, 3) Computing work done by variable forces, 4) Solving differential equations, 5) Analyzing accumulation functions, 6) Computing probabilities in statistics, 7) Solving physics problems involving rates of change.

Question 9

How do you handle integration constants?

Accepted Answer

In indefinite integrals, always add +C (the integration constant) because the derivative of any constant is zero. In definite integrals, the constant cancels out: ∫f(x)dx from a to b = [F(x) + C]ₐᵇ = [F(b) + C] - [F(a) + C] = F(b) - F(a). The constant doesn't affect the final result in definite integrals.

Question 10

What are common mistakes when using FTC?

Accepted Answer

Common FTC mistakes include: 1) Forgetting to find the antiderivative first, 2) Not applying the bounds correctly in F(b) - F(a), 3) Confusing definite and indefinite integrals, 4) Forgetting the +C in indefinite integrals, 5) Not checking that the function is continuous on the interval, 6) Using the wrong antiderivative (any antiderivative works in FTC Part 2).

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