Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Fundamental Theorem of Calculus: Definite Integral Evaluation
**Problem:**
Use the Fundamental Theorem to evaluate the definite integral exactly.
\[ \int_{0}^{5} t^3 \, dt = \]
**Solution:**
To evaluate the integral \(\int_{0}^{5} t^3 \, dt\), apply the Fundamental Theorem of Calculus, which states that if \(F\) is an antiderivative of \(f\) on an interval \([a, b]\), then:
\[ \int_{a}^{b} f(t) \, dt = F(b) - F(a) \]
For the given integral, we need to find the antiderivative of \(t^3\):
1. The antiderivative of \(t^3\) is \( \frac{t^4}{4} + C \), where \(C\) is a constant of integration.
2. Evaluate this antiderivative at the bounds 5 and 0.
\[ \left[ \frac{t^4}{4} \right]_{0}^{5} = \left( \frac{5^4}{4} \right) - \left( \frac{0^4}{4} \right) \]
\[ = \frac{625}{4} - 0 = \frac{625}{4} \]
Thus, the exact value of the definite integral is:
\[ \int_{0}^{5} t^3 \, dt = \frac{625}{4} \]
This solution demonstrates the step-by-step application of the Fundamental Theorem of Calculus to evaluate the given definite integral.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3190a26b-07a4-4acc-b1dd-52ddc1659ccb%2Fbae3c2be-0a63-4f42-826a-85fe4ce0e96e%2Fstjxfg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Fundamental Theorem of Calculus: Definite Integral Evaluation
**Problem:**
Use the Fundamental Theorem to evaluate the definite integral exactly.
\[ \int_{0}^{5} t^3 \, dt = \]
**Solution:**
To evaluate the integral \(\int_{0}^{5} t^3 \, dt\), apply the Fundamental Theorem of Calculus, which states that if \(F\) is an antiderivative of \(f\) on an interval \([a, b]\), then:
\[ \int_{a}^{b} f(t) \, dt = F(b) - F(a) \]
For the given integral, we need to find the antiderivative of \(t^3\):
1. The antiderivative of \(t^3\) is \( \frac{t^4}{4} + C \), where \(C\) is a constant of integration.
2. Evaluate this antiderivative at the bounds 5 and 0.
\[ \left[ \frac{t^4}{4} \right]_{0}^{5} = \left( \frac{5^4}{4} \right) - \left( \frac{0^4}{4} \right) \]
\[ = \frac{625}{4} - 0 = \frac{625}{4} \]
Thus, the exact value of the definite integral is:
\[ \int_{0}^{5} t^3 \, dt = \frac{625}{4} \]
This solution demonstrates the step-by-step application of the Fundamental Theorem of Calculus to evaluate the given definite integral.
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