To determine
a)
To evaluate:
The given integral by interpreting it in terms of thearea.
Answer
4
Explanation
Concept:
A definite integral can be interpreted as a net area, that is, difference of the areas ∫abf(x)dx=A1-A2
where A1 is the area of region above x- axis and below the graph of f, and A2 is the area of region below x- axis and above the graph of f.
Given:
∫02f(x)dx
Calculations:
To find the area under the curve, from the points 0 to 2, means to calculate the area of the shaded region as shown in the figure below.
From the graph, A1 is the area of atrapezoid.
The formula for area of trapezoid is
A=12·(b1+b2)·h
Here,b1=1, b2=3, and h=2
A1=12·(b1+b2)·h=12·(1+3)·2=4 square units
Conclusion:
∫02f(x)dx=4
To determine
b)
To evaluate:
The given integral by interpreting it in terms of area
Answer
10
Explanation
Concept:
A definite integral can be interpreted as a net area, that is, difference of the areas ∫abf(x)dx=A1-A2
where A1 is the area of region above x- axis and below the graph of f, and A2 is the area of region below x- axis and above the graph of f.
Given:
∫05f(x)dx
Calculations:
To find the area under the curve, from the points 0 to 5 means, to calculate the area of the shaded region as shown in the figure below.
Now, from the previous we have
A1=4 square units
A2 is area of a rectangle.
The area of a rectangle is
A=l·b
Here, length = 1 units and breadth = 3 units so
A2=l·b=1·3=3 square units
A3 is the area of a triangle.
The formula for the area of a triangle is
A=12·b·h
Here, base = 2 units and height = 3 units
A3=12·b·h=12·2·3=3 square units
Therefore, net area is sum of three areas (because all areas are above x=axis).
∫05f(x)dx=A1+A2+A3=4+3+3=10 square units
Conclusion:
∫02f(x)dx=10
To determine
c)
To evaluate:
The given integral by interpreting it in terms of area.
Answer
-3
Explanation
Concept:
A definite integral can be interpreted as a net area, that is, difference of the areas ∫abf(x)dx=A1-A2
where A1 is the area of region above x- axis and below the graph of f, and A2 is the area of region below x- axis and above the graph of f.
Given:
∫57f(x)dx
Calculations:
To find the area under the curve from the points 5 to 7, means to calculate the area of the shaded region as shown in the figure below.
From the graph, A1 is the area of a triangle.
The formula for the area of a triangle is.
A=12·b·h
Here, base = 2 units and height = 3 units
A1=12·b·h=12·2·3=3 square units
Since the area lies below x axis, the area will be negative.
Therefore, the net area is
∫57f(x)dx=-A1=-3 square units
Conclusion:
∫57f(x)dx=-3
To determine
d)
To evaluate:
The given integral by interpreting it in terms of area.
Answer
2
Explanation
Concept:
A definite integral can be interpreted as a net area, that is, difference of the areas ∫abf(x)dx=A1-A2
where A1 is the area of region above x- axis and below the graph of f, and A2 is the area of region below x- axis and above the graph of f.
Given:
∫09f(x)dx
Calculations:
To find the area under the curve from the points 0 to 9 means to calculate the area of the shaded region as shown in below figure,
Now, from the previous solution, we have
A1=4, A2=3, A3=3, A4=-3
From the graph, A5 is the area of a trapezoid.
The formula for the area of a trapezoid is
A=12·(b1+b2)·h
Here, b1=3, b2=2 and h=2
A5=12·(b1+b2)·h=12·(3+2)·2=5 square units
Since the area lies below the x axis, the area will be negative.
Therefore, the net area is
∫09f(x)dx=A1+A2+A3-A4-A5=2 square units
Conclusion:
∫09f(x)dx=2