Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter7: Integration
Section7.3: Area And The Definite Integral
Problem 1E: Explain the difference between an indefinite integral and a definite integral.
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Evaluate the
=______________+ C
![The image shows a mathematical integral. Here is the transcription suitable for an educational website:
Evaluate the following definite integral:
\[ \int_{1}^{4} \sqrt{x} \, dx \]
This integral represents the area under the curve \( \sqrt{x} \) from \(x = 1\) to \(x = 4\). To solve this, remember to:
1. Find the antiderivative of \(\sqrt{x}\).
2. Evaluate it from the lower limit of 1 to the upper limit of 4.
Mathematically, this can be expressed and calculated as follows:
1. Rewrite the integrand: \(\sqrt{x} = x^{1/2}\).
2. Find the antiderivative: \(\int x^{1/2} \, dx = \frac{2}{3} x^{3/2} + C\).
3. Apply the Fundamental Theorem of Calculus to evaluate the definite integral:
\[ \left[ \frac{2}{3} x^{3/2} \right]_{1}^{4} \]
By substituting the upper and lower limits:
\[ \frac{2}{3} (4)^{3/2} - \frac{2}{3} (1)^{3/2} \]
Simplify and calculate the result:
\[ \frac{2}{3} (8) - \frac{2}{3} (1) = \frac{16}{3} - \frac{2}{3} = \frac{14}{3} \]
So, the value of the integral is \(\boxed{\frac{14}{3}}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F44f9ebf7-05a0-44a3-89d0-aa50ed2840a0%2Fb2e2064d-cf8e-48a1-823a-21d2d88009af%2Ffeprrzq.jpeg&w=3840&q=75)
Transcribed Image Text:The image shows a mathematical integral. Here is the transcription suitable for an educational website:
Evaluate the following definite integral:
\[ \int_{1}^{4} \sqrt{x} \, dx \]
This integral represents the area under the curve \( \sqrt{x} \) from \(x = 1\) to \(x = 4\). To solve this, remember to:
1. Find the antiderivative of \(\sqrt{x}\).
2. Evaluate it from the lower limit of 1 to the upper limit of 4.
Mathematically, this can be expressed and calculated as follows:
1. Rewrite the integrand: \(\sqrt{x} = x^{1/2}\).
2. Find the antiderivative: \(\int x^{1/2} \, dx = \frac{2}{3} x^{3/2} + C\).
3. Apply the Fundamental Theorem of Calculus to evaluate the definite integral:
\[ \left[ \frac{2}{3} x^{3/2} \right]_{1}^{4} \]
By substituting the upper and lower limits:
\[ \frac{2}{3} (4)^{3/2} - \frac{2}{3} (1)^{3/2} \]
Simplify and calculate the result:
\[ \frac{2}{3} (8) - \frac{2}{3} (1) = \frac{16}{3} - \frac{2}{3} = \frac{14}{3} \]
So, the value of the integral is \(\boxed{\frac{14}{3}}\).
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