Question

Find the area bounded by the axis of x, the curve y = 2x2 and the tangent to the curve at
the point whose abscissa is 2.​

Answer 1

Step-by-step explanation:

Given Find the area bounded by the axis of x, the curve y = 2x2 and the tangent to the curve at  the point whose abscissa is 2.

• The given curve is an upward parabola with vertex at (0,0)
• Now if x = 2
• So y = 2x^2
• Or y = 2(2)^2
• Or y = 8
• Therefore the point of the curve will be P(2,8)
• Differentiating with respect to x we get dy / dx = 2.2x = 4x
• Therefore the slope of tangent to curve is y – y1 = m(x – x1)
• So slope at P(2,8) is y – 8 = 8(x – 2)
•                            So y = 8x – 8
• It meets x axis y = 0 at 8x – 8 = 0
•                                Or x = 1
• Now area of shaded region is equal to required area.
• = 0 to 2 ʃ2x^2 dx – 1 to 2ʃ(8x – 8) dx
• = 2 (x^3 / 3)0 to 2 – (8 . x^2/2 – 8x) 1 to 2
• = 2/3(8 – 0) – [16 – 16) – (4 – 8)]
• = 16 / 3 – 4
• = 4/3 sq units