13.04 Properties of definite integrals
Evaluate the following definite integrals:
\int_{0}^{3} \left( 4 x + 5\right) dx
\int_{ - 2 }^{0} \left( 10 x + 4\right) dx
\int_{2}^{4} \left( 6 x + 5\right) dx
\int_{3}^{4} \left( - 4 x + 3\right) dx
\int_{ - 4 }^{5} \left( - 8 x + 3\right) dx
\int_{ - 10 }^{ - 6 } \left( - 5 x + 8\right) dx
\int_{ - 2 }^{0} 9 x^{2} dx
\int_{ - 1 }^{3} 9 x^{2} dx
\int_{ - 2 }^{1} \left(x^{2} + 4\right) dx
\int_{ - 1 }^{2} \left( 9 x^{2} + 1\right) dx
\int_{4}^{6} \left( 9 x^{2} + 2 x + 7\right) dx
\int_{ - 4 }^{2} x \left(x - 4\right) dx
\int_{0}^{4} 5 x^{\frac{3}{2}} dx
\int_{ - 3 }^{6} x \left(x + 3\right) \left(x - 6\right) dx
\int_{6}^{11} \sqrt{x - 2} dx
\int_{3}^{6} \left(\sqrt{x - 2} + 5\right) dx
\int_{3}^{4} \left( 2 x + 3\right)^{3} dx
\int_{ - 4 }^{2} \left(\left(x + 2\right)^{3} + 3\right) dx
\int_{1}^{2} \dfrac{x^{5} - x^{ - 2 }}{x^{2}} dx
Consider the function f \left( x \right) = x^2. Find the value of the following:
Consider the function f \left( x \right) = 2 x.
Find the value of \int_{0}^{3} f \left( x \right) dx.
Find the value of \int_{0}^{3} 5f \left( x \right) dx.
Find the value of 5\int_{0}^{3} f \left( x \right)dx.
State the property of definite integrals demonstrated by parts (b) and (c).
Given that \int_{ - 2 }^{7} f \left( x \right) dx = 2, find \int_{ - 2 }^{7} 5 f \left( x \right) dx.
Given that \int_{ - 1 }^{6} f \left( x \right) dx = 3, find \int_{ - 1 }^{6} \left( 9 f \left( x \right) - 2\right) dx.
Consider the function f \left( x \right) = 6 x.
Find the value of \int_{4}^{8} f \left( x \right) dx.
Find the value of \int_{8}^{4} f \left( x \right) dx.
State the property of definite integrals demonstrated by parts (a) and (b).
Given that \int_{4}^{6} f \left( x \right) dx = 3, find the values of the following:
\int_{6}^{4} f \left( x \right) dx
\int_{4}^{6} 3 f \left( x \right) dx
\int_{4}^{6} \left(f \left( x \right) + x\right) dx
Given that \int_{ - 1 }^{2} f \left( x \right) dx = 4 and \int_{2}^{8} f \left( x \right) dx = 8, find the values of the folowing:
\int_{ - 1 }^{8} f \left( x \right) dx
\int_{8}^{ - 1 } f \left( x \right) dx
\int_{ - 1 }^{2} 2 f \left( x \right) dx + \int_{2}^{8} 3 f \left( x \right) dx
Given that \int_{ - 1 }^{3} f \left( x \right) dx = 5 and \int_{2}^{3} f \left( x \right) dx = 2, find the values of the following:
\int_{ - 1 }^{2} f \left( x \right) dx
\int_{3}^{ - 1 } f \left( x \right) dx
2 \int_{ - 1 }^{2} f \left( x \right) dx + \int_{2}^{3} 3 f \left( x \right) dx
Consider the graph of y = f \left( x \right).
Find the value of \int_{0}^{2} f \left( x \right) dx.
Find the value of \int_{2}^{5} f \left( x \right) dx.
Find the value of \int_{5}^{6} f \left( x \right) dx.
Hence state the area bounded by the function and the x-axis.
Write a single definite integral to represent the area bounded by the function and the x-axis.
Consider the graph of y = f \left( x \right).
Find the value of \int_{0}^{4} f \left( x \right) dx.
Find the value of \int_{4}^{6} f \left( x \right) dx.
Find the value of \int_{6}^{8} f \left( x \right) dx.
Hence state the area bounded by the function and the x-axis.
Write a single definite integral to represent the area bounded by the function and the x-axis.
Consider the graph of y = f \left( x \right).
Find the value of \int_{0}^{5} f \left( x \right) dx.
Find the value of \int_{5}^{7} f \left( x \right) dx.
Hence find the value of \int_{0}^{7} f \left( x \right) dx.
State the area bounded by the function and the x-axis.
Write a single definite integral to represent the area bounded by the function and the x-axis.
Consider the graph of y = f \left( x \right).
Find the value of \int_{0}^{4} f \left( x \right) dx.
Find the exact value of \int_{4}^{16} f \left( x \right) dx.
Hence find the exact value of \int_{0}^{16} f \left( x \right) dx.
Calculate the exact area bounded by the curve, the x-axis and the y-axis.
Write an expression for the exact area from part (d) as a sum or difference of definite integrals.
Consider the function y = -5.
State whether the graph is above or below the x-axis.
Calculate \int_{ - 4 }^{3} \left(-5\right) \ dx.
Hence find the area bounded by the curve, the x-axis and the bounds x = - 4 and x = 3.
Consider the function y = 2 x - 8.
Find the x-intercept of the function.
State the values of x for which the graph is above the x-axis.
Calculate \int_{ - 2 }^{4} \left( 2 x - 8\right) dx.
Hence find the area bounded by the line, the x-axis and the bounds x = - 2 and x = 4.
Consider the function y = \left(x - 3\right) \left(x - 9\right).
Find the x-intercepts of the function.
State the values of x for which the graph is below the x-axis.
Calculate \int_{3}^{9} \left(x - 3\right) \left(x - 9\right) dx.
Hence find the area bounded by the curve, the x-axis and the bounds x = 3 and x = 9.
Consider the function y = - \left(x + 2\right) \left(x + 8\right).
Find the x-intercepts of the function.
State the values of x for which the graph is above the x-axis.
Calculate \int_{ - 8 }^{ - 2 } - \left(x + 2\right) \left(x + 8\right) dx.
Hence find the area bounded by the curve, the x-axis and the bounds x = - 8 and \\x = - 2.
Consider the graph of the curve y = x^{2} + 6.
Find the exact area of the shaded region.
Consider the graph of the curve y = 6 x^{2}.
Find the exact area of the shaded region.
Consider the graph of the line x + y = 3.
Find the exact area of the shaded region.
Consider the graph of the curve y = 4 - x^{2}.
Find the exact area of the shaded region.
Consider the graph of the curve \\y = x \left(x - 1\right) \left(x + 3\right).
Find the exact area of the shaded region.
Consider the given functions.
Sketch a graph of the function.
Hence calculate the exact area bounded by the curve, x = 1, x = 3 and the x-axis.
y = 2 x + 3
y = - 2 x + 8
y = 4x- x^2 - 3
Consider the function y = \left(x - 1\right)^{2} \left(x + 3\right).
Sketch a graph of the function.
Hence calculate the exact area bounded by the curve, x-axis and the lines x = - 1 and x = 2.
Consider the function y = \left(x + 1\right)^{3} + 2\left(x + 1\right)^{3} + 2.
Sketch a graph of the function.
Hence calculate the exact area bounded by the curve, x-axis, and the lines x = - 2 and x = 1.
Consider the function y = \sqrt{x + 1}.
Sketch a graph of the function.
Hence calculate the exact area bounded by the curve, the x-axis, and the line x = 3.
For each of the following functions:
Sketch a graph of the function.
Hence determine the exact area bounded by the curve and the x-axis.
y = x \left(x - 1\right) \left(x + 3\right)
y = \left(x - 1\right) \left(x + 2\right) \left(x + 3\right)
y = x \left(x - 1\right)
Consider the function f \left( x \right) as shown. The numbers in the shaded regions indicate the area of the region.
Find the value of the following:
\int_{ - 3 }^{0} f \left( x \right) dx
\int_{ - 3 }^{2} f \left( x \right) dx
The area enclosed by the curve and the x-axis for x < 0.
The area enclosed by the curve and the x-axis.
Consider the graph of y = f \left( x \right).
Find the value of \int_{0}^{3} f \left( x \right) dx.
Find the value of \int_{3}^{8} f \left( x \right) dx.
Hence calculate \int_{0}^{8} f \left( x \right) dx.
Calculate the area bounded by the function, the x-axis and the y-axis.
Write an expression for the area from part (d) as a sum or difference of definite integrals.
Write an expression for the area from part (d) using an absolute value sign.
Consider the graph of y = f \left( x \right).
Find the value of \int_{0}^{8} f \left( x \right) dx.
Calculate the area bounded by the function, the x-axis and the y-axis.
Write an expression for the area from part (b) as a sum or difference of definite integrals.
Write an expression for the area from part (b) using an absolute value sign.
Consider the function f \left( x \right) as shown. The numbers in the shaded regions indicate the area of the region.
Find the following:
\int_{ - 5 }^{3} f \left( x \right) dx
\left|\int_{ - 5 }^{3} f \left( x \right) dx\right|
\int_{ - 5 }^{3} \left|f \left( x \right)\right| dx
The area enclosed by the curve and the x-axis.
Consider the function f \left( x \right) as shown. The numbers in the shaded regions indicate the area of the region.
Find the following:
\int_{ - 2 }^{7} f \left( x \right) dx
\int_{3}^{7} \left( - f \left( x \right) \right) dx
\int_{ - 2 }^{7} 2 f \left( x \right) dx
\int_{3}^{ - 2 } f \left( x \right) dx + \int_{3}^{7} f \left( x \right) dx
\int_{ - 2 }^{7} \left|f \left( x \right)\right| dx
The area enclosed by the curve and the x-axis.
Consider the function f \left( x \right) as shown. The numbers in the shaded regions indicate the area of the region.
Find the following:
\int_{ - 4 }^{0} f \left( x \right) dx
\left|\int_{ - 4 }^{3} f \left( x \right) dx\right|
\int_{ - 1 }^{0} 2 f \left( x \right) dx + \int_{3}^{0} f \left( x \right) dx
The area enclosed by the curve and the x-axis.
\int_{ - 4 }^{3} \left| f \left( x \right)\right| dx
\int_{ - 4 }^{3} \left(f \left( x \right) + x^{2}\right) dx given that the definite integral \int_{ - 4 }^{3} x^{2} dx = \dfrac{91}{3}.
Consider the function f \left( x \right) as shown. The numbers in the shaded regions indicate the area of the region.
Write the following in terms of A and B:
\int_{ - 5 }^{1} f \left( x \right) dx
\int_{ - 5 }^{ - 2 } 3 f \left( x \right) dx - \int_{ - 2 }^{1} f \left( x \right) dx
\left|\int_{ - 5 }^{1} f \left( x \right) dx\right|
\int_{ - 5 }^{1} \left|f \left( x \right)\right| dx
\int_{ - 5 }^{1} \left(f \left( x \right) + x\right) dx given that the definite integral \int_{ - 5 }^{1} x \ dx = - 12.
Consider the function f \left( x \right) as shown. The numbers in the shaded regions indicate the area of the region.
Write the following in terms of A, B and C:
\int_{0}^{6} f \left( x \right) dx
\int_{6}^{ - 1 } f \left( x \right) dx
The area bounded by the curve and the x-axis.
\int_{ - 1 }^{6} \left|f \left( x \right)\right| dx
\int_{ - 1 }^{6} \left( 2 x - f \left( x \right)\right) dx given that the definite integral \int_{ - 1 }^{6} 2 x \ dx = 35
Consider the function f \left( x \right) where x = - 4, 1 and 3 are the only x-intercepts and \int_{ - 4 }^{1} f \left( x \right) dx = 4 and \int_{1}^{3} f \left( x \right) dx = - 7.
Find the following:
\int_{ - 4 }^{3} f \left( x \right) dx
\int_{ - 4 }^{3} \left|f \left( x \right)\right| dx of f \left( x \right).
The area bounded by the curve of f \left( x \right) and the x-axis.
\int_{3}^{ - 4 } \left(f \left( x \right) - x^{3}\right) dx, given that \int_{3}^{ - 4 } x^{3} dx = \dfrac{175}{4}.
Consider the function f \left( x \right) where x = - 2, 2 and 8 are the only x-intercepts and \int_{ - 2 }^{2} f \left( x \right) dx = - 5 and \int_{ - 2 }^{8} f \left( x \right) dx = 3.
Find the following:
\int_{2}^{8} f \left( x \right) dx
The area bounded by the curve and the x-axis.
\int_{ - 2 }^{8} \left( 2 f \left( x \right) - 6 x\right) \ dx, given that \int_{ - 2 }^{8} x \ dx = 30.
Consider the function f \left( x \right) where x= - 6, - 2, 2 and 7 are the only x-intercepts and \int_{ - 6 }^{ - 2 } f \left( x \right) dx = - A , \int_{ - 2 }^{2} f \left( x \right) dx = B and \int_{2}^{7} f \left( x \right) dx = - C.
Write the following in terms of A, B and C:
\left|\int_{ - 6 }^{7} f \left( x \right) dx\right|
The area bounded by the curve and the x-axis.
\int_{ - 2 }^{7} f \left( x \right) dx - \int_{ - 6 }^{2} f \left( x \right) dx
\int_{ - 6 }^{2} 2 f \left( x \right) dx + \int_{2}^{7} \dfrac{f \left( x \right)}{2} dx
\int_{ - 6 }^{7} \left(3 - f \left( x \right) + x\right) dx, given that \int_{ - 6 }^{7} \left(3 + x\right) dx = \dfrac{91}{2}.