Question
Download Solution PDFComprehension
The slope of the tangent to the curve y = f(x) at (x, f(x) ) is 4 for every real number x and the curve passes through the origin.
. What is the area bounded by the curve, the x-axis and the line x = 4?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFCalculation:
Given,
The equation of the curve is y = 4x , and the line x = 4 intersects the curve at the point (4, 16) . We need to find the area bounded by the curve, the x-axis, and the line x = 4 .
The region of interest is a right triangle with a base along the x-axis from x = 0 to x = 4 and a height of 16 units, corresponding to the point (4, 16) .
The area of the triangle is given by the formula:
\( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \)
Substituting the values of the base (4 units) and the height (16 units):
\( \text{Area} = \frac{1}{2} \times 4 \times 16 = 32 \, \text{square units} \)
∴ The area is 32 square units.
Hence, the correct answer is option 3.
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