Area Under the Curve Question - None

Question

If a curve $y = ax + bx$ passes through point $(1, 2)$ and the area bounded by curve, line $x = 4$ and x-axis is 8, then:

(a) $a = 3$

(b) $b = 3$

(c) $a = -1$

(d) $b = -1$

Accepted Answer

Solution: The curve is $y = ax + bx$ (likely $y = a\sqrt{x} + bx$ based on context).

Condition 1: Passes through $(1, 2)$:

$2 = a(1) + b(1) = a + b$

Condition 2: Area bounded by curve, line $x = 4$ and x-axis is 8:

$\int_0^4 (a\sqrt{x} + bx)dx = 8$

$\left[\frac{2a}{3}x^{3/2} + \frac{b}{2}x^2\right]_0^4 = 8$

$\frac{2a}{3}(8) + \frac{b}{2}(16) = 8$

$\frac{16a}{3} + 8b = 8$

From $a + b = 2$: $a = 2 - b$

$\frac{16(2-b)}{3} + 8b = 8$

$\frac{32 - 16b}{3} + 8b = 8$

$32 - 16b + 24b = 24$

$8b = -8$ ⟹ $b = -1$

$a = 2 - (-1) = 3$

∴ Answer is (a, d)

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