Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 5 - Integration

Problema 5.R.102c

Capitolo 5, Problema 5.R.102c

Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.
(c) Evaluate H '(2) .

Guida verificata passo dopo passo

Step 1: Recognize that the function H(𝓍) is defined as an integral, H(𝓍) = ∫₀ˣ √(4 − t²) dt. To find H'(𝓍), we use the Fundamental Theorem of Calculus, which states that if F(𝓍) = ∫ₐˣ f(t) dt, then F'(𝓍) = f(𝓍), provided f is continuous.

Step 2: Apply the Fundamental Theorem of Calculus to H(𝓍). This gives H'(𝓍) = √(4 − 𝓍²), because the integrand √(4 − t²) is continuous for the given domain.

Step 3: Substitute 𝓍 = 2 into the derivative H'(𝓍). This means H'(2) = √(4 − 2²).

Step 4: Simplify the expression inside the square root. Compute 4 − 2², which simplifies to 4 − 4.

Step 5: Conclude that H'(2) = √(0). The derivative at this point is determined by evaluating the square root of the simplified expression.

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Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then the integral of f from a to b can be computed as F(b) - F(a). This theorem also implies that if H(x) is defined as an integral of a function, then H'(x) can be found by evaluating the integrand at the upper limit of integration.

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Fundamental Theorem of Calculus Part 1

Differentiation of an Integral Function

When differentiating a function defined by an integral, such as H(x) = ∫₀ˣ f(t) dt, the derivative H'(x) can be computed using the integrand evaluated at the upper limit. Specifically, H'(x) = f(x), provided that f is continuous on the interval. This principle simplifies the process of finding derivatives of integral-defined functions.

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Integrals Resulting in Basic Trig Functions Example 1

Evaluating the Integrand

To evaluate H'(2) in the given problem, we first need to identify the integrand, which is √(4 - t²). We then substitute the upper limit of integration, x = 2, into the integrand. This step is crucial as it allows us to find the value of the derivative at that specific point, which is essential for solving the problem.

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Completing the Square to Rewrite the Integrand

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