Textbook Question
Area functions from graphs The graph of Ζ is given in the figure. A(π) = β«βΛ£ Ζ(t) dt and evaluate A(2), A(5), A(8), and A(12).
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8. Definite Integrals
Fundamental Theorem of Calculus
1:35 minutesProblem 5.3.101b
Textbook Question
{Use of Tech} Functions defined by integrals Consider the function g, which is given in terms of a definite integral with a variable upper limit.
b) Calculate g'(π)
g(π) = β«βΛ£ sinΒ² t dt
Verified step by step guidance1
Step 1: Recognize that the function g(π) is defined as a definite integral with a variable upper limit. This is a classic application of the Fundamental Theorem of Calculus, which states that if g(π) = β«βΛ£ f(t) dt, then g'(π) = f(π), provided f is continuous.
Step 2: Identify the integrand in the given function g(π). Here, the integrand is sinΒ²(t). This means that g(π) = β«βΛ£ sinΒ²(t) dt.
Step 3: Apply the Fundamental Theorem of Calculus. According to the theorem, the derivative of g(π) with respect to π is simply the integrand evaluated at the upper limit of integration. Therefore, g'(π) = sinΒ²(π).
Step 4: Confirm that the lower limit of integration (0 in this case) does not affect the derivative because it is a constant. The derivative depends only on the variable upper limit and the integrand.
Step 5: Write the final expression for g'(π). Based on the steps above, g'(π) = sinΒ²(π). This completes the differentiation process.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus connects differentiation and integration, stating that if a function is defined as an integral with a variable upper limit, its derivative can be found by evaluating the integrand at that upper limit. Specifically, if g(x) = β«βΛ£ f(t) dt, then g'(x) = f(x). This theorem is essential for calculating the derivative of functions defined by integrals.
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Definite Integral
A definite integral represents the accumulation of quantities, such as area under a curve, between two specified limits. In the context of the function g(x) = β«βΛ£ sinΒ² t dt, the integral calculates the area under the curve of sinΒ² t from 0 to x. Understanding how to evaluate definite integrals is crucial for applying the Fundamental Theorem of Calculus.
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Chain Rule
The Chain Rule is a fundamental principle in calculus used to differentiate composite functions. When applying the Chain Rule, if a function g(x) is defined in terms of another function, such as an integral, the derivative g'(x) can be found by multiplying the derivative of the outer function by the derivative of the inner function. This concept is particularly relevant when dealing with integrals that have variable limits.
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