Area Under the Curve
Area with implicitly defined functions
Grade 12
Question:
<p>The area of the region bounded by the curve <span class="math">y = f(x)</span>, the x-axis, and the lines <span class="math">x = a</span> and <span class="math">x = b</span>, where <span class="math">-\infty < a < b < -2</span>, is</p>
<p>(A) <span class="math">\int_a^b \frac{x^2}{3(f(x))^2 - 1} dx + bf(b) - af(a)</span></p>
<p>(B) <span class="math">-\int_a^b \frac{x^2}{3(f(x))^2 - 1} dx + bf(b) - af(a)</span></p>
<p>(C) <span class="math">\int_a^b \frac{x^2}{3(f(x))^2 - 1} dx - bf(b) + af(a)</span></p>
<p>(D) <span class="math">-\int_a^b \frac{x^2}{3(f(x))^2 - 1} dx - bf(b) + af(a)</span></p>
Step-by-Step Solution
Key Concept: The integral represents an area calculation involving an implicitly defined function; integration by parts with the given constraint yields the formula.
<p>Use integration by parts and the implicit relation defining <span class="math">f(x)</span> to express the area in terms of the given integral formula.</p>
Correct Answer: B