In order to find the area of the region determined by the curve y = x + 1 along with the lines x = 2, x = 3 and the x-axis, we first need to set up the definite integral. Step 1: Set up the integral To calculate the area, we will integrate the function y = x + 1 with respect to x between the limitsRead more
In order to find the area of the region determined by the curve y = x + 1 along with the lines x = 2, x = 3 and the x-axis, we first need to set up the definite integral.
Step 1: Set up the integral
To calculate the area, we will integrate the function y = x + 1 with respect to x between the limits x = 2 and x = 3.
A = ∫₂³ (x + 1) dx
Step 2: Integrate the function
First, integrate (x + 1):
∫ (x + 1) dx = (x²)/2 + x
Now, evaluate this from x = 2 to x = 3:
At x = 3:
(3²)/2 + 3 = 9/2 + 3 = 9/2 + 6/2 = 15/2
At x = 2:
(2²)/2 + 2 = 4/2 + 2 = 2 + 2 = 4
Step 3: Find the area
A= 15/2 – 4 = 15/2 – 8/2 = 7/2
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 inRead more
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:
To find the area bounded by the curve y = sin x, the x-axis, and the ordinates x = 0 and x = 2, we must calculate the definite integral of sin x from 0 to 2. Step 1: Set up the integral The area is given by: A = ∫₀² sin x dx Step 2: Solve the integral We know the integral of sin x is: ∫ sin x dx = -Read more
To find the area bounded by the curve y = sin x, the x-axis, and the ordinates x = 0 and x = 2, we must calculate the definite integral of sin x from 0 to 2.
Step 1: Set up the integral
The area is given by:
A = ∫₀² sin x dx
Step 2: Solve the integral
We know the integral of sin x is:
∫ sin x dx = -cos x
Now, evaluate the integral from 0 to 2:
A = [-cos x]₀²
At x = 2:
-cos(2)
At x = 0:
-cos(0) = -1
Thus, the area is:
A = -cos(2) – (-1) = 1 + cos(2)
Step 3: Approximate the result
Using a calculator, cos(2) ≈ -0.416, so:
A ≈ 1 – (-0.416) = 1 + 0.416 = 1.416
Thus, the closest option to this result is: 4/3 sq. units.
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 ∫Read more
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 =
The area bounded by the curve y = log x, the x-axis, and the ordinates x = 1 and x = 2 can be found using the definite integral of log x from x = 1 to x = 2. Step 1: Write the integral The area is calculated as: A = ∫₁² log x dx Step 2: Integrate This can be evaluated as a result of integration by pRead more
The area bounded by the curve y = log x, the x-axis, and the ordinates x = 1 and x = 2 can be found using the definite integral of log x from x = 1 to x = 2.
Step 1: Write the integral
The area is calculated as:
A = ∫₁² log x dx
Step 2: Integrate
This can be evaluated as a result of integration by parts. Note that the integration by parts formula is as follows:
∫ u dv = uv – ∫ v du
Here let u = log x and dv = dx, then du = (1/x) dx, v = x
Thus
∫ log x dx = x log x – ∫ x (1/x) dx = x log x – x
Step 3: Calculate the result
Now integrate the integral from x = 1 to x = 2:
The area of the region bounded by the curve y = x + 1 and the lines x = 2, x = 3 and x-axis in sq. units is
In order to find the area of the region determined by the curve y = x + 1 along with the lines x = 2, x = 3 and the x-axis, we first need to set up the definite integral. Step 1: Set up the integral To calculate the area, we will integrate the function y = x + 1 with respect to x between the limitsRead more
In order to find the area of the region determined by the curve y = x + 1 along with the lines x = 2, x = 3 and the x-axis, we first need to set up the definite integral.
Step 1: Set up the integral
To calculate the area, we will integrate the function y = x + 1 with respect to x between the limits x = 2 and x = 3.
A = ∫₂³ (x + 1) dx
Step 2: Integrate the function
First, integrate (x + 1):
∫ (x + 1) dx = (x²)/2 + x
Now, evaluate this from x = 2 to x = 3:
At x = 3:
(3²)/2 + 3 = 9/2 + 3 = 9/2 + 6/2 = 15/2
At x = 2:
(2²)/2 + 2 = 4/2 + 2 = 2 + 2 = 4
Step 3: Find the area
A= 15/2 – 4 = 15/2 – 8/2 = 7/2
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The area of curve xy = 4, bounded by the lines x =1 and x = 3 and x-axis in sq. units is
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 inRead more
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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he area bounded by the curve y = sin x, x – axis, ordinates x = 0 and x = 2x is
To find the area bounded by the curve y = sin x, the x-axis, and the ordinates x = 0 and x = 2, we must calculate the definite integral of sin x from 0 to 2. Step 1: Set up the integral The area is given by: A = ∫₀² sin x dx Step 2: Solve the integral We know the integral of sin x is: ∫ sin x dx = -Read more
To find the area bounded by the curve y = sin x, the x-axis, and the ordinates x = 0 and x = 2, we must calculate the definite integral of sin x from 0 to 2.
Step 1: Set up the integral
The area is given by:
A = ∫₀² sin x dx
Step 2: Solve the integral
We know the integral of sin x is:
∫ sin x dx = -cos x
Now, evaluate the integral from 0 to 2:
A = [-cos x]₀²
At x = 2:
-cos(2)
At x = 0:
-cos(0) = -1
Thus, the area is:
A = -cos(2) – (-1) = 1 + cos(2)
Step 3: Approximate the result
Using a calculator, cos(2) ≈ -0.416, so:
A ≈ 1 – (-0.416) = 1 + 0.416 = 1.416
Thus, the closest option to this result is: 4/3 sq. units.
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The area bounded by the curve y = sin x, x-axis, ordinates x = 0 and x = 2π is
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 ∫Read more
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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The area bounded by the curve y log x, x-axis, ordinates x = 1 and x = 2 in sq. units is
The area bounded by the curve y = log x, the x-axis, and the ordinates x = 1 and x = 2 can be found using the definite integral of log x from x = 1 to x = 2. Step 1: Write the integral The area is calculated as: A = ∫₁² log x dx Step 2: Integrate This can be evaluated as a result of integration by pRead more
The area bounded by the curve y = log x, the x-axis, and the ordinates x = 1 and x = 2 can be found using the definite integral of log x from x = 1 to x = 2.
Step 1: Write the integral
The area is calculated as:
A = ∫₁² log x dx
Step 2: Integrate
This can be evaluated as a result of integration by parts. Note that the integration by parts formula is as follows:
∫ u dv = uv – ∫ v du
Here let u = log x and dv = dx, then du = (1/x) dx, v = x
Thus
∫ log x dx = x log x – ∫ x (1/x) dx = x log x – x
Step 3: Calculate the result
Now integrate the integral from x = 1 to x = 2:
A = [x log x – x]₁²
When x = 2,
2 log 2 – 2
When x = 1,
1 log 1 – 1 = 0 – 1 = -1
Hence area is:
A = (2 log 2 – 2) – (-1) = 2 log 2 – 2 + 1 = 2 log 2 – 1
Now substituting the value for log 2 = 0.693.
A ≈ 2(0.693) – 1 = 1.386 – 1 = 0.386
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