Business Calculus: Definite Integrals and Applications (Fundamental Theorem of Calculus)

Q1. Evaluate the definite integral $\int_{-1}^{3} (x+4)\,dx$.

Background

Topic: Definite Integrals & Fundamental Theorem of Calculus

This question tests your ability to compute a definite integral using antiderivatives and the limits of integration.

Key Terms and Formulas

• Definite Integral: $\int_{a}^{b} f(x)\,dx$ gives the net area under $f(x)$ from $x=a$ to $x=b$.

• Fundamental Theorem of Calculus: If $F(x)$ is an antiderivative of $f(x)$, then $\int_{a}^{b} f(x)\,dx = F(b) - F(a)$.

Step-by-Step Guidance

1. Find the antiderivative $F(x)$ of $f(x) = x+4$. Recall that the antiderivative of $x$ is $\frac{1}{2}x^2$ and of $4$is $4x$.

2. Write $F(x) = \frac{1}{2}x^2 + 4x$.

3. Apply the Fundamental Theorem: $\int_{-1}^{3} (x+4)\,dx = F(3) - F(-1)$.

4. Compute $F(3)$ and $F(-1)$ using your antiderivative.

Try solving on your own before revealing the answer!

Q2. Evaluate the definite integral $\int_{1}^{4} (x - \frac{1}{2})\,dx$.

Background

Topic: Definite Integrals & Antiderivatives

This question asks you to integrate a linear function over a given interval.

Key Terms and Formulas

• Antiderivative: The function whose derivative is the given function.

• Definite Integral: $\int_{a}^{b} f(x)\,dx = F(b) - F(a)$, where $F(x)$ is an antiderivative of $f(x)$.

Step-by-Step Guidance

1. Find the antiderivative of $x - \frac{1}{2}$. The antiderivative of $x$ is $\frac{1}{2}x^2$, and of $-\frac{1}{2}$ is $-\frac{1}{2}x$.

2. Write $F(x) = \frac{1}{2}x^2 - \frac{1}{2}x$.

3. Apply the limits: $\int_{1}^{4} (x - \frac{1}{2})\,dx = F(4) - F(1)$.

4. Substitute $x=4$ and $x=1$ into $F(x)$ and find the difference.

Try solving on your own before revealing the answer!

Q3. Evaluate the definite integral $\int_{1}^{2} (t^6 - t^2)\,dt$.

Background

Topic: Definite Integrals of Polynomial Functions

This question tests your ability to integrate polynomials and apply the limits of integration.

Key Terms and Formulas

• Power Rule for Integration: $\int x^n dx = \frac{1}{n+1}x^{n+1} + C$ (for $n \neq -1$)

• Definite Integral: $\int_{a}^{b} f(x) dx = F(b) - F(a)$

Step-by-Step Guidance

1. Find the antiderivative of $t^6$ and $-t^2$ separately.

2. Combine the antiderivatives: $F(t) = \frac{1}{7}t^7 - \frac{1}{3}t^3$.

3. Apply the limits: $F(2) - F(1)$.

4. Substitute $t=2$ and $t=1$ into $F(t)$ and find the difference.

Try solving on your own before revealing the answer!

Q4. Evaluate the definite integral $\int_{0}^{1} 4x^3\,dx$.

Background

Topic: Definite Integrals of Monomials

This question asks you to integrate a monomial function and evaluate it over a given interval.

Key Terms and Formulas

• Power Rule for Integration: $\int x^n dx = \frac{1}{n+1}x^{n+1} + C$

• Definite Integral: $\int_{a}^{b} f(x) dx = F(b) - F(a)$

Step-by-Step Guidance

1. Find the antiderivative of $4x^3$. Use the power rule: $\int 4x^3 dx = 4 \cdot \frac{1}{4}x^4 = x^4$.

2. Apply the limits: $x=1$ and $x=0$.

3. Compute $F(1) - F(0)$, where $F(x) = x^4$.

Try solving on your own before revealing the answer!

Q5. Evaluate the definite integral $\int_{0}^{1} (1 + x^4)^6 dx$.

Background

Topic: Substitution in Definite Integrals

This question tests your ability to use substitution (u-substitution) to evaluate a definite integral.

Key Terms and Formulas

• u-Substitution: Let $u = 1 + x^4$, then $du = 4x^3 dx$.

• Adjust the limits of integration when changing variables.

Step-by-Step Guidance

1. Let $u = 1 + x^4$. Compute $du = 4x^3 dx$.

2. Rewrite the integral in terms of $u$ and $du$.

3. Change the limits of integration: when $x=0$, $u=1$; when $x=1$, $u=2$.

4. Integrate $u^6 du$ over the new limits.

Try solving on your own before revealing the answer!

Q6. Evaluate the definite integral $\int_{1}^{5} e^{5x} dx$.

Background

Topic: Exponential Functions and Definite Integrals

This question tests your ability to integrate exponential functions and apply the limits of integration.

Key Terms and Formulas

• Integral of $e^{ax}$: $\int e^{ax} dx = \frac{1}{a}e^{ax} + C$

• Definite Integral: $\int_{a}^{b} f(x) dx = F(b) - F(a)$

Step-by-Step Guidance

1. Find the antiderivative of $e^{5x}$, which is $\frac{1}{5}e^{5x}$.

2. Apply the limits: $x=5$ and $x=1$.

3. Compute $F(5) - F(1)$, where $F(x) = \frac{1}{5}e^{5x}$.

Try solving on your own before revealing the answer!

Q7. A certain object moves so that its velocity (in m/s) after time $t$ (in s) is $v = t^2 + 4t + 6$. Find the distance traveled during the first four seconds by evaluating $\int_{0}^{4} (t^2 + 4t + 6) dt$.

Background

Topic: Applications of Definite Integrals (Distance from Velocity)

This question asks you to find the total distance traveled by integrating the velocity function over a time interval.

Key Terms and Formulas

• Distance Traveled: $\int_{a}^{b} v(t) dt$ gives the net change in position (displacement) over $[a, b]$.

• Antiderivative of a Polynomial: Use the power rule for each term.

Step-by-Step Guidance

1. Find the antiderivative of $t^2 + 4t + 6$.

2. Write $F(t) = \frac{1}{3}t^3 + 2t^2 + 6t$.

3. Apply the limits: $F(4) - F(0)$.

4. Substitute $t=4$ and $t=0$ into $F(t)$ and find the difference.

Try solving on your own before revealing the answer!

Q8. After a new firm starts in business, it finds that the rate of profit (in hundreds of dollars) after $t$ years is $P'(t) = 6t^2 + 4t + 8$. Find the profit in year 4 of operation.

Background

Topic: Business Applications of Definite Integrals (Accumulated Profit)

This question asks you to find the total profit accumulated over a period by integrating the rate of profit function.

Key Terms and Formulas

• Accumulated Profit: $\int_{0}^{4} P'(t) dt$ gives the total profit over the first 4 years.

• Antiderivative of a Polynomial: Use the power rule for each term.

Step-by-Step Guidance

1. Find the antiderivative of $6t^2 + 4t + 8$.

2. Write $P(t) = 2t^3 + 2t^2 + 8t$.

3. Apply the limits: $P(4) - P(0)$.

4. Substitute $t=4$ and $t=0$ into $P(t)$ and find the difference. Remember to multiply your final result by 100 to convert from hundreds of dollars to dollars.

Try solving on your own before revealing the answer!