Calculus Study Guide: Area Between Curves and Limits

Q11. Find the area of the shaded region between $y = x^4 - 32$ and $y = -x^4$.

Background

Topic: Area Between Curves (Definite Integrals)

This question tests your ability to set up and evaluate a definite integral to find the area between two curves. You need to determine the points of intersection and integrate the difference between the upper and lower functions.

Key Terms and Formulas:

• Area between curves: $A = \int_{a}^{b} [f(x) - g(x)] \, dx$

• Points of intersection: Solve $x^4 - 32 = -x^4$ for $x$.

Step-by-Step Guidance

1. Set the two equations equal to each other to find the points of intersection: $x^4 - 32 = -x^4$.

2. Solve for $x$ to determine the limits of integration.

3. Identify which function is on top (greater value) and which is on the bottom within the interval.

4. Set up the integral for the area: $A = \int_{a}^{b} [(x^4 - 32) - (-x^4)] \, dx$.

5. Simplify the integrand before integrating.

Try solving on your own before revealing the answer!

Final Answer:

The area is $\int_{-2}^{2} [2x^4 - 32] \, dx$.

After evaluating the definite integral, you get the total area of the shaded region.

Q12. Find the area of the shaded region between $y = -x^4$ and $y = x^2 - 2x$.

Background

Topic: Area Between Curves (Definite Integrals)

This question asks you to find the area between two curves, which involves setting up a definite integral using the points where the curves intersect.

Key Terms and Formulas:

• Area between curves: $A = \int_{a}^{b} [f(x) - g(x)] \, dx$

• Points of intersection: Solve $-x^4 = x^2 - 2x$ for $x$.

Step-by-Step Guidance

1. Set the two equations equal to each other to find the intersection points: $-x^4 = x^2 - 2x$.

2. Solve for $x$ to determine the limits of integration.

3. Determine which function is above the other in the interval.

4. Set up the integral: $A = \int_{a}^{b} [(x^2 - 2x) - (-x^4)] \, dx$.

5. Simplify the integrand before integrating.

Try solving on your own before revealing the answer!

Final Answer:

The area is $\int_{0}^{2} [x^2 - 2x + x^4] \, dx$.

Evaluating this definite integral gives the area of the shaded region.

Q14. Use the graph to estimate the limit of the function.

Background

Topic: Limits from Graphs

This question tests your ability to estimate the value of a function as $x$ approaches a certain point from the left and right, using a graph. You are asked to find the left-hand limit, right-hand limit, and the overall limit at $x = \frac{\pi}{2}$.

Key Terms and Formulas:

• Left-hand limit: $\lim_{x \to c^-} f(x)$

• Right-hand limit: $\lim_{x \to c^+} f(x)$

• Limit: $\lim_{x \to c} f(x)$

Step-by-Step Guidance

1. Examine the graph at $x = \frac{\pi}{2}$ and observe the values as $x$ approaches from the left and right.

2. Identify the value the function approaches from the left ($x \to \frac{\pi}{2}^-$).

3. Identify the value the function approaches from the right ($x \to \frac{\pi}{2}^+$).

4. Compare the left and right limits to determine if the overall limit exists at $x = \frac{\pi}{2}$.

Try solving on your own before revealing the answer!

Final Answer:

The left-hand limit is $\lim_{x \to \frac{\pi}{2}^-} f(x) = 3$, the right-hand limit is $\lim_{x \to \frac{\pi}{2}^+} f(x) = 5$, and the overall limit does not exist since the left and right limits are not equal.