Download App

Areas of Bounded Regions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsAreas of Bounded Regions5 Marks
Question
Compare the area under the curve $\text{y}=\cos^2\text{x}\text{ and }\text{y}=\sin^2\text{x}$ between x = 0 and $\text{x}=\pi.$
✓

Answer


Consider the value of y for different values of x
$\text{x}$ $0$ $\frac{\pi}{4}$ $\frac{\pi}{3}$ $\frac{\pi}{2}$ $\frac{2\pi}{3}$ $\frac{5\pi}{6}$ $\pi$
$\text{y}=\cos^2\text{x}$ $1$ $0.5$ $ 0.25$ $0$ $ 0.25$ $ 0.75$ $1$
$$$\text{y}=\sin^2\text{x}$ $0$ $0.5$ $ 0.75$ $1$ $ 0.75$ $0.25$ $0$
Let $A_1$ be the area of curve $\text{y}=\cos^2\text{x}$ between x = 0 and $\text{x}=\pi$
Let $ A_2$ be the area of curve $\text{y}=\sin^2\text{x}$ between x = 0 and $\text{x}=\pi$
Consider, a vertical strip of lenght = |y| and width = $\text{dx}$ in the shaded region of both the curves The area of approximating $=|\text{y}|\text{dx}$
The approximating rectangle moves from x = 0 to $\text{x}=\pi$
$\text{A}_1=\int\limits^\pi_0\text{|y| dx}$
$[0\leq\text{x}\leq\pi,\text{ y}>0\Rightarrow|\text{y}|=\text{y}]$
$\Rightarrow\text{A}_1=\int\limits^\pi_0\cos^2\text{x dx}$
$\Rightarrow\text{A}_1=\int\limits^\pi_0\text{y dx}$
$\Rightarrow\text{A}_1=\int\limits^\pi_0(1+\cos2\text{x})\text{dx}$
$\Rightarrow\text{A}_1=\frac{1}{2}\big[\text{x}+\frac{\sin2\text{x}}{2}\big]^\pi_0$
$\Rightarrow\text{A}_1=\frac{1}{2}\big[\text{x}+\frac{\sin2\text{x}}{2}-0\big]$
$\Rightarrow\text{A}_1=\frac{\pi}{2}\text{sq. units}$
Also,
$\text{A}_2=\int\limits^\pi_0\text{|y| dx}$
$\Rightarrow\text{A}_2=\int\limits^\pi_0\text{y dx}$
$\Rightarrow\text{A}_2=\int\limits^\pi_0\sin^2\text{x dx}$
$\Rightarrow\text{A}_2=\bigg[\frac{x}{2}-\frac{1}{2}\frac{\sin2\text{}x}{2}\bigg]^\pi_0$
$\Rightarrow\text{A}_2=\frac{x}{2}-\bigg(\frac{1}{2}\frac{\sin2\pi}{2}\bigg)$
$\Rightarrow\text{A}_2=\frac{\pi}{2}\text{sq. units}$
$\therefore$ Area of curves $\text{y}=\cos^2\text{x}$ and area of curve $\text{y}=\sin^2\text{x}$ are both equal to $\frac{\pi}{2}\text{sq. units}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Areas of Bounded Regions→Areas of Bounded Regions — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Verify Rolle's theorem for the following function on the indicated intervals$\text{f}(\text{x})=\cos2\Big(\text{x}-\frac{\pi}{4}\Big)\text{ on }\Big[0,\frac{\pi}{2}\Big]$
→
Evaluate $\int _ { - 1 } ^ { 2 } \left| x ^ { 3 } - x \right| d x$.
→
Evaluate the following definite integrals:
$\int_{0}^\limits{\infty}\frac{1}{\text{a}^2+\text{b}^2\text{x}^2} \text{ dx}$
→
Sketch the graph of y = | x + 3 | and evaluate the area under the curve y = | x + 3 | above x-axis and between x = – 6 to x = 0.
→
Show that the following systems of linear equations is inconsistent:
3x - y + 2z = 3,
2x + y + 3z = 5,
x - 2y - z = 1
→
Evaluate the following integrals:
$\int\limits^{\pi}_0\text{x}\cos^2\text{x dx}$
→
An insurance company insured $3000$ scooters, $4000$ cars and $5000$ trucks. The probabilities of the accident involving a scooter, a car and a truck are $0.02, 0.03$ and $0.04$ respectively. One of the insured vehicles meet with an accident. Find the probability that it is a,
  1. Scooter.
  2. Car.
  3. Truck.
→
Evaluate: $\int\limits_0^{\pi/2} 2 \sin x \cos x \tan^{-1}(\sin x) dx.$
→
The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after t seconds.
→
In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
$4x + 8y + z - 8 = 0$ and $y + z - 4 = 0$
→