Determine the area of the region bounded by the curve xy = 4 and the lines x = 1, x = 3, and the x-axis. First we rewrite the equation xy = 4 to solve for y in terms of x.

Now solving the above equation for y we get,
y = 4/x

Step 1
We begin by setting up the integral
The area can be calculated by integrating y = 4/x from x = 1 to x = 3:

A = ∫₁³ (4/x) dx

Step 2: Evaluate the integral
We know that:

∫ (1/x) dx = ln |x|

Thus, the integral becomes:

A = 4 ∫₁³ (1/x) dx = 4 [ln x]₁³

Step 3: Calculate the area
At x = 3:

ln 3

At x = 1:

ln 1 = 0

Therefore, the area is

A = 4 (ln 3 – 0) = 4 ln 3

Using logarithmic properties:

A = ln (3⁴) = ln 81

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We have to calculate the definite integral of sin x from 0 to 2π in order to find the area bounded by the curve y = sin x, the x-axis, and the ordinates x = 0 and x = 2π.

Step 1: Setup the integral
The area is given by:
A = ∫₀²π sin x dx

Step 2: Solve the integral
We know the integral of sin x is

∫ sin x dx = -cos x

Evaluate the integral from 0 to 2π:

A = [-cos x]₀²π

At x = 2π:

-cos(2π) = -1

At x = 0:

-cos(0) = -1

Hence, area is:

A = -1 – (-1) = 0

Since sin x is above the x-axis for the interval [0, π] and below the x-axis for the interval [π, 2π], the areas of these two parts are equal in magnitude but opposite in sign. So we take the absolute value of the integrals over both intervals.

Step 3: Evaluate the area with absolute value
Area =

A = 2 × ∫₀π sin x dx = 2 [-cos x]₀π

At x = π:

-cos(π) = 1

At x = 0:

-cos(0) = -1

Thus, the area is: A = 2 × (1 – (-1)) = 2 × 2 = 4

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