Question

Area of the region bounded by the curve y=✓49-x^2 and the x-axis is ?​

Answer 1

Answer:

The area of the region bounded by the curve $y=\sqrt{49-x^{2} }$ and the x-axis is 24.5π sq. units.

Step-by-step explanation:

Given:

Equation of curve $y=\sqrt{49-x^{2} }$

To find:

Area of the bounded region (A)

Step 1:

We know that the equation of the x-axis is given as $y=0$.

Substituting this in the given equation of the curve.

$\sqrt{49-x^{2} } = 0$

Squaring both sides, we get

$49-x^{2} =0$

$x^{2} =49$

x = ±7

Therefore, the points of intersection of the given lines are (7, 0) and (-7, 0).

Step 2:

To find the area of the region, we integrate the curve equation between the points of intersection. Therefore,

$A = \int\limits^7_{-7} {\sqrt{49-x^{2} } } \, dx$

$A= [\frac{x}{2}\sqrt{7^{2}-x^{2} } + \frac{7^{2} }{2} sin^{-1} \frac{x}{7} ]^{7}_{-7}$

$A=[\frac{7}{2}\sqrt{7^{2} -7^{2} } +\frac{7^{2} }{2}sin^{-1} \frac{7}{7} ]-[\frac{-7}{2}\sqrt{7^{2} -(-7)^{2} } +\frac{(-7)^{2} }{2}sin^{-1} \frac{-7}{7}]$

$A=[0+\frac{49}{2}sin^{-1} 1 ]-[0+\frac{49}{2}sin^{-1} (-1)]$

$A=\frac{49}{2}\frac{\pi }{2}-[-\frac{49}{2}\frac{\pi }{2}]$

$A=\frac{49}{4}\pi + \frac{49}{4}\pi$

$A=\frac{98}{4}\pi$

$A=24.5\pi sq. units$

Therefore, the area bounded is 24.5π sq. units.

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Answer 2

The area of the region bounded by the curve and the x-axis would be 24.5π square units

Given

• y=✓(49-x^2)

To find

• Area of the region

Solution

we are provided with the equation of a curve and are asked to estimate the area of the region which is bounded by the curve and the x-axis.

the equation of the curve,

y=✓(49-x^2)

squaring both sides,

y^2 = 49 - x^2

or, x^2 + y^2 = 49

or, x^2 + y^2 = 7^2

Therefore the given equation of the curve gets simplified into the equation of a circle with origin as the centre,

we know that the circle with origin as the centre is symmetric about x-axis.

therefore the area will be equally divided by the x-axis.

area of the circle = πr^2

r = 7

Therefore, the area of the circle above the x-axis would be

1/2 π(7^2)

1/2 π(49)

or, 24.5 π square units.

Hence, the area of the region bounded by the curve and the x-axis would be 24.5π square units

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