12.07 Properties of definite integrals
Given that \int_{ - 2 }^{7} f \left( x \right)\, dx = 2, evaluate \int_{ - 2 }^{7} 5 f \left( x \right)\, dx.
Consider the function f \left( x \right) where \int_{ - 1 }^{6} f \left( x \right)\, dx = 3. Determine \int_{ - 1 }^{6} \left( 9 f \left( x \right) - 2\right)\, dx.
Suppose \int_{4}^{6} f \left( x \right)\, dx = 3. Find the value of:
\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
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.
What property of definite integrals do parts (a) and (b) demonstrate?
Consider the even function y = 3 x^{4}.
Evaluate the integral \int_{0}^{1} 3 x^{4}\, dx.
Hence, find \int_{ - 1 }^{1} 3 x^{4}\, dx.
The graph of y = f \left( x \right) has been drawn:
Find the value of a, where a \gt 0, so that \int_{ - a }^{a} f \left( x \right) \, dx = 0.
Suppose \int_{ - 1 }^{2} f \left( x \right)\, dx = 4 and \int_{2}^{8} f \left( x \right)\, dx = 8. Find the value of:
\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
Suppose \int_{ - 1 }^{3} f \left( x \right)\, dx = 5 and \int_{2}^{3} f \left( x \right)\, dx = 2. Find the value of:
\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 function f \left( x \right) where \int_{ - 4 }^{1} f \left( x \right)\, dx = 4 and \int_{1}^{3} f \left( x \right)\, dx = - 7.
Determine \int_{ - 4 }^{3} f \left( x \right)\, dx.
If - 4, 1 and 3 are the only x-intercepts of f \left( x \right), determine \int_{ - 4 }^{3} \left|f \left( x \right)\right|\, dx.
Determine the area bounded by the curve of f \left( x \right) and the x-axis.
Determine \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 \int_{ - 2 }^{2} f \left( x \right)\, dx = - 5 and \int_{ - 2 }^{8} f \left( x \right)\, dx = 3.
Determine \int_{2}^{8} f \left( x \right)\, dx.
If - 2, 2 and 8 are the only x-intercepts of f \left( x \right), determine the area bounded by the curve f \left( x \right) and the x-axis.
Determine \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 \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 expressions for each of the following in terms of A, B and C:
\left|\int_{ - 6 }^{7} f \left( x \right)\, dx\right|
\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}.
The area bounded by the curve f \left( x \right) and the x-axis if - 6, - 2, 2 and 7 are the only x-intercepts of f \left( x \right).
Consider the odd function y = x^{3}.
Evaluate the integral \int_{0}^{2} x^{3}\, dx.
Hence, write down the answer to \int_{ - 2 }^{2} x^{3}\, dx.
Write down the area bounded by the function y = x^{3}, the x-axis and the lines x = - 2 and x = 2.
The diagram shows the region bounded by y = \dfrac{1}{x + 3}, x = 0, x = 45 and y = 0:
The region is divided into two parts of equal area, by the line x = k where k > 0.
Find the value of k.
Consider the function f \left( x \right) shown. The numbers inside the shaded regions indicate the area of the region:
Determine \int_{ - 3 }^{0} f \left( x \right)\, dx.
Determine the area enclosed by the curve and the x-axis for x < 0.
Determine \int_{ - 3 }^{2} f \left( x \right)\, dx.
Determine the total area enclosed by the curve and the x-axis.
Find the exact area of the shaded region bounded by the line x + y = 3, the x-axis, the y-axis, and the line x=6 shown:
Find the exact area of the shaded region bounded by the curve y = 4 - x^{2}, the y-axis, the x-axis and the line x=-3:
Find the exact area of the shaded region between the curve y = x \left(x - 1\right) \left(x + 3\right) and the x-axis:
Consider the function y = \left(x - 1\right) \left(x + 2\right) \left(x + 3\right).
Sketch the graph of the function.
Hence, determine the exact area bounded by the curve and the x-axis.
Consider the function y = x \left(x - 1\right) \left(x + 3\right).
Sketch the graph of the function.
Hence, determine the exact area bounded by the curve and the x-axis.
Consider the function f \left( x \right) shown below. The numbers in the shaded regions indicate the area of the region:
Find \int_{ - 5 }^{3} f \left( x \right)\, dx.
Find the area enclosed by the curve f \left( x \right) and the x-axis.
Find \left|\int_{ - 5 }^{3} f \left( x \right)\, dx\right|.
Find \int_{ - 5 }^{3} \left|f \left( x \right)\right|\, dx.
Consider the function f \left( x \right) shown below. The numbers in the shaded regions indicate the area of the region:
Find \int_{ - 2 }^{7} f \left( x \right)\, dx.
Find the area enclosed by the curve f \left( x \right) and the x-axis.
Find \int_{3}^{7} \left( - f \left( x \right) \right)\, dx.
Find \int_{ - 2 }^{7} 2 f \left( x \right)\, dx.
Find \int_{3}^{ - 2 } f \left( x \right)\, dx + \int_{3}^{7} f \left( x \right)\, dx.
Find \int_{ - 2 }^{7} \left|f \left( x \right)\right|\, dx.
Consider the function f \left( x \right) shown below. The numbers in the shaded regions indicate the area of the region:
Find \int_{ - 4 }^{0} f \left( x \right)\, dx.
Find the area enclosed by the curve f \left( x \right) and the x-axis.
Find \left|\int_{ - 4 }^{3} f \left( x \right)\, dx\right|.
Find \int_{ - 1 }^{0} 2 f \left( x \right)\, dx + \int_{3}^{0} f \left( x \right)\, dx.
Find \int_{ - 4 }^{3} \left(f \left( x \right) + x^{2}\right)\, dx, given that \int_{ - 4 }^{3} x^{2}\, dx = \dfrac{91}{3}.
Consider the function f \left( x \right) shown below. The letters in the shaded regions represent 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 \\ \int_{ - 5 }^{1} x\, dx = - 12
Consider the function f \left( x \right) shown below. The letters 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 f \left( x \right) 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 \int_{ - 1 }^{6} 2 x \, dx = 35
Consider the function y = f \left( x \right) shown:
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.
Find the exact area bounded by the function y=f(x), the x-axis and the y-axis.
Consider the function y = f \left( x \right) shown:
Find \int_{0}^{3} f \left( x \right)\, dx.
Find \int_{3}^{8} f \left( x \right)\, dx.
Hence, find \int_{0}^{8} f \left( x \right)\, dx.
Find the area bounded by the function, the x-axis and the y-axis.
Consider the function y = f \left( x \right) shown:
Find \int_{0}^{8} f \left( x \right)\, dx.
Calculate the area bounded by the function, the x-axis and the y-axis.
Consider the function y = f \left( x \right) shown:
Which integral has the greatest value?
Consider the graph of y = f \left( x \right):
The area of region R is 2\text{ units}^2, and the area of region S is 3\text{ units}^2. It is given that \int_{0}^{4} f \left( x \right) \, dx = 10.
Find \int_{ - 1 }^{3} f \left( x \right) \, dx.
