Pregunta
MT2-2
a)
AmDCrath
Answer any 1 questions from 5 to 6 . Each carries 4 Marks.
5 a) Find the area enclosed by the circle
using integration.
b) If
, then
.
6. a) Find the area of the region bounded by
and the
-axis in the first quadrant.
b) Find
a) Find
-
(3)
(1)
b)
किणनाइक
Answer any 2 questions from 7 to 9 . Each carries 6 Marks .
7. a) The length
of a rectangle is increasing at the rate of
and the width
is decreasing at the rate of
. When
and
.
i) Find the rates of change of its area.
ii) Find the rates of change of the perimeter of the rectangle.
a)
5 a) Find the area enclosed by the circle
b) If
6. a) Find the area of the region bounded by
b) Find
a) Find
(3)
(1)
b)
7. a) The length
i) Find the rates of change of its area.
ii) Find the rates of change of the perimeter of the rectangle.
Ask by Rodriquez Ruiz. in India
Jan 08,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
Solución
To evaluate the integral:
we can use a substitution method involving inverse trigonometric functions.
Step 1: Recognize the Integral Form
The integral resembles the standard form:
where
and
.
Step 2: Apply Substitution
Let’s set:
Substituting these into the integral:
Simplify inside the square root:
So the integral becomes:
Step 3: Integrate
The integral of
is
:
Step 4: Substitute Back
Recall that
:
Final Answer
Where
is the constant of integration.
Respondido por UpStudy AI y revisado por un tutor profesional
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Extra Insights
To solve the integral
, we can use a trigonometric substitution. Let
. Consequently,
, and the integral transforms nicely into
after adjusting the limits based on the substitution. This ultimately leads to
.
Now, for the area enclosed by the circle given by the equation
, we can set up the integral by solving for
:
. The area is found using the symmetry of the circle:
. Evaluating this integral with a substitution (like
) is a fun challenge that ultimately gives you the total area of
square units!

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