11.04 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
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 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 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