Problem 7
Question
Compute (note: change \(x\) to \(t\) if it confuses you.) a. \(\int_{0}^{1}\left[3+x^{2}\right] d x\) b. \(\int_{1}^{2} 3 x^{2} d x\) c. \(\int_{3}^{5} 3 x^{3}-6 x^{2} d x\)
Step-by-Step Solution
Verified Answer
a. \( \frac{10}{3} \), b. 7, c. \( \frac{577}{4} \)
1Step 1: Understanding the First Integral
For the first integral, \( \int_{0}^{1} [3+x^2] \, dx \), we need to integrate the function \( 3+x^2 \) with respect to \( x \) from 0 to 1. We will break this task into two parts: integrating \( 3 \) and \( x^2 \).
2Step 2: Integrate Constant Function
The integral of a constant, \( 3 \), with respect to \( x \) is \( 3x \). Therefore, when evaluating from 0 to 1, it becomes \( 3(1) - 3(0) = 3 \).
3Step 3: Integrate Power Function
The integral of \( x^2 \) with respect to \( x \) is \( \frac{x^3}{3} \). Evaluating from 0 to 1, it becomes \( \frac{1^3}{3} - \frac{0^3}{3} = \frac{1}{3} \).
4Step 4: Sum Results of First Integral
Add the results from Steps 2 and 3 for the first integral: \( 3 + \frac{1}{3} = \frac{10}{3} \).
5Step 5: Understanding the Second Integral
For \( \int_{1}^{2} 3x^2 \, dx \), we need to integrate the function \( 3x^2 \) with respect to \( x \) from 1 to 2.
6Step 6: Compute Second Integral
The integral of \( 3x^2 \) is \( x^3 \). Evaluating this from 1 to 2 gives us \( 2^3 - 1^3 = 8 - 1 = 7 \).
7Step 7: Understanding the Third Integral
For the integral \( \int_{3}^{5} (3x^3 - 6x^2) \, dx \), we need to integrate the function \( 3x^3 - 6x^2 \) with respect to \( x \) from 3 to 5.
8Step 8: Integrate Each Term Separately
Integrate \( 3x^3 \) to get \( \frac{3x^4}{4} \) and \( -6x^2 \) to get \( -2x^3 \).
9Step 9: Evaluate Integrals for Third Integral
Evaluate \( \frac{3x^4}{4} - 2x^3 \) from 3 to 5. This evaluates to: \[ \left(\frac{3(5)^4}{4} - 2(5)^3\right) - \left(\frac{3(3)^4}{4} - 2(3)^3\right) = (\frac{1875}{4} - 250) - (\frac{243}{4} - 54) = \frac{577}{4} \].
Key Concepts
Definite IntegralsIntegration TechniquesPower Rule for IntegrationComputing Areas Under a Curve
Definite Integrals
Definite integrals are a fundamental concept in integral calculus, used to compute the accumulated value of a function over a specific interval. They are represented in the form \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the limits of integration. This notation tells us to integrate the function \( f(x) \) from \( x = a \) to \( x = b \).
This process calculates the net area under the curve of \( f(x) \) from \( a \) to \( b \). The result is a number that gives insight into the total accumulation, such as distance traveled or total growth over a time period.
One crucial aspect is that while the definite integral considers the function's area under the curve, it can also account for negative areas, where the function dips below the x-axis. Integrating over these regions requires careful evaluation to ensure we obtain the correct net area.
This process calculates the net area under the curve of \( f(x) \) from \( a \) to \( b \). The result is a number that gives insight into the total accumulation, such as distance traveled or total growth over a time period.
One crucial aspect is that while the definite integral considers the function's area under the curve, it can also account for negative areas, where the function dips below the x-axis. Integrating over these regions requires careful evaluation to ensure we obtain the correct net area.
Integration Techniques
Integration techniques provide various approaches for solving integrals, especially when dealing with complex functions. Each technique has appropriate scenarios where it simplifies the integration process.
Common techniques include:
Common techniques include:
- Substitution: Useful for integrals that resemble the derivative of a composition of functions.
- Integration by Parts: Ideal for products of functions, using the formula \( \int u \, dv = uv - \int v \, du \).
- Partial Fractions: Applies to rational functions to break them into simpler fractions.
Power Rule for Integration
The power rule for integration is one of the most straightforward and fundamental techniques in calculus. It simplifies the process of integrating polynomials by providing a direct way to find the antiderivative.
The power rule states that the integral of \( x^n \) with respect to \( x \) is \( \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \) and \( C \) is the constant of integration. This rule easily transforms polynomial functions into their antiderivatives.
For example, in the integration of \( x^2 \), the power rule tells us to raise the exponent by one, yielding \( x^3 \), and then divide by the new exponent, giving \( \frac{x^3}{3} \).
This method was used in each part of the original exercise to integrate terms like \( 3 \) (as \( 3x \)) and \( x^2 \) (as \( \frac{x^3}{3} \)). Understanding the power rule is crucial for efficiently tackling basic integrals.
The power rule states that the integral of \( x^n \) with respect to \( x \) is \( \frac{x^{n+1}}{n+1} + C \), where \( n eq -1 \) and \( C \) is the constant of integration. This rule easily transforms polynomial functions into their antiderivatives.
For example, in the integration of \( x^2 \), the power rule tells us to raise the exponent by one, yielding \( x^3 \), and then divide by the new exponent, giving \( \frac{x^3}{3} \).
This method was used in each part of the original exercise to integrate terms like \( 3 \) (as \( 3x \)) and \( x^2 \) (as \( \frac{x^3}{3} \)). Understanding the power rule is crucial for efficiently tackling basic integrals.
Computing Areas Under a Curve
Computing areas under a curve is a primary application of definite integrals in calculus, offering a practical way to measure "total quantity" over an interval. This can represent various real-world scenarios, such as total revenue, distance, or even energy.
To compute this area, you set up the definite integral \( \int_{a}^{b} f(x) \, dx \), where \( f(x) \) is the curve, and \( a \), \( b \) are the boundaries. The process of evaluating this integral allows you to capture the net area, considering both positive and negative regions relative to the x-axis.
In the exercise, each computed integral provides the net area under different polynomial functions over specified intervals. The solutions integrate the polynomial expressions and evaluate them exactly at the defined limits to obtain the result.
By understanding how definite integrals work and applying the power rule, students can accurately calculate these areas, gaining insights into the total effect of a function over a given range.
To compute this area, you set up the definite integral \( \int_{a}^{b} f(x) \, dx \), where \( f(x) \) is the curve, and \( a \), \( b \) are the boundaries. The process of evaluating this integral allows you to capture the net area, considering both positive and negative regions relative to the x-axis.
In the exercise, each computed integral provides the net area under different polynomial functions over specified intervals. The solutions integrate the polynomial expressions and evaluate them exactly at the defined limits to obtain the result.
By understanding how definite integrals work and applying the power rule, students can accurately calculate these areas, gaining insights into the total effect of a function over a given range.
Other exercises in this chapter
Problem 6
Is the logarithm function, \(L(x)=\ln (x)\), linear? Prove or disprove.
View solution Problem 6
Approximate A. \(\int_{0}^{1} e^{t} d t\) B. \(\int_{0}^{\pi} \sin (t) d t\)
View solution Problem 7
Use Definition of Integral II to evaluate $$\int_{1}^{2} \frac{1}{t^{2}} d t$$ Partition [1,2] in \(n\) equal subintervals by $$t_{0}=1, \quad t_{1}=1+\frac{1}{
View solution Problem 8
a. Compute: \(\mathrm{P}_{1} \quad \int_{2}^{4}\left[t \times t^{2}\right] d t\) and $$\mathrm{P}_{2} \int_{2}^{4} t d t \times \int_{2}^{4} t^{2} d t$$ b. Comp
View solution