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Question #41478

Difficulty: 8/10 | 🟢 Free

The Question

Let $m$ and $M$ be the global minimum and global maximum values of the function $f(x) = x^3 - 3x^2 + 2$ on the interval $[-1, 4]$ . Let $A(t)$ be the area of the region in the $xy$ -plane bounded by the curve $y = 2x^3 + 2x - 2$ , the $y$ -axis, and the horizontal line $y = t$ (for $t \ge -2$ ). Find the value of $15 \int_{m}^{M} (A'(t))^2 \, dt$

Options

A.
656
B.
28
C.
628
D.
420

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Axis of Integration Selection|Leibniz Rule for Differentiation under Integral|Global Extrema on Intervals

Why This Question Is Tricky

Common Mistakes

This problem tests the intersection of global extrema analysis and advanced integration techniques.

The primary trap lies in identifying the correct global extrema; students often merely check critical points ( $x=0, 2$ ) and incorrectly conclude $M=2$ , leading to the trap answer 28.

Furthermore, finding $A'(t)$ requires either applying the Leibniz Rule to an x-axis integral (where the boundary evaluation vanishes) or astutely switching the axis of integration to the y-axis to utilize the First Fundamental Theorem of Calculus.

Failing to correctly parameterize the moving upper bound $h^{-1}(t)$ derails the entire differentiation process.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Axis of Integration Selection
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free