Question

The area of the region bounded by $y=x^3$, x-axis, $x=-2$ and $x=4$ is

• A. 64
• B. 81/4
• C. 66/5
• D. 68

Hint:

When calculating the area bounded by a curve and the x-axis, you must use the absolute value of the function, $|f(x)|$. This often requires splitting the integral into multiple intervals at the points where the function crosses the x-axis (i.e., at its roots).

Answer 1

The Correct Option is D

To find the area of the region bounded by the curve \( y = x^3 \), the x-axis, and the vertical lines \( x = -2 \) and \( x = 4 \), we need to evaluate the definite integral of the function over the given interval.

The area \( A \) is given by the integral:

\(A = \int_{-2}^{4} x^3 \, dx\)

Let's compute the integral step-by-step:

1. The antiderivative of \( x^3 \) is \( \frac{x^4}{4} \).
2. Evaluate the definite integral:
3. Substitute the upper limit \( x = 4 \):
4. Substitute the lower limit \( x = -2 \):
5. Calculate the area by subtracting the lower limit evaluation from the upper limit evaluation:

However, due to our initial condition where we evaluate the region between the graph and the x-axis, we notice that because \( x^3 \) is negative in part of this interval (specifically from \(-2\) to \(0\)), we must take the absolute value to consider the total area.

1. Since \( y = x^3 \) is below the x-axis for \( x \in [-2, 0] \) and above for \( x \in [0, 4] \), we have:
2. Compute each part separately:
   • Total area from \(-2\) to \(0\):
   • Total area from \(0\) to \(4\):
3. Sum of both parts:

Therefore, the total area of the region is 68. The correct answer is 68.