Find the exact value of the following definite integrals using the graphs provided:
\int_{1}^{8} \left(10 - x\right) \,\, dx
\int_{3}^{5} \left( 2 x + 1\right) \, dx
\int_{ - 6 }^{6} \sqrt{36 - x^{2}} \, dx
For each of the following graphs, find the exact value of \int_{0}^{16} f \left( x \right) dx:
Consider the function y = f \left( x \right):
Find the value of:
Hence, calculate the area bounded by the function and the x-axis.
Evaluate the following:
Evaluate the following:
Find the exact area of the shaded region under the curve y = 6 x^{2}.
Find the exact area of the shaded region under the curve y = x^{2} + 6.
Consider the function y = x \left(x - 1\right).
Sketch the graph of the function.
Hence, determine the exact area bounded by the curve and the x-axis.
Consider the function y = \dfrac{1}{x - 2} + 3.
Sketch the graph of the function.
Hence, determine the exact area bounded by the curve, the x-axis, and the lines x = 4 and x = 6.
Consider the function y = \sqrt{x + 1}.
Sketch the graph of the function.
Hence, determine the exact area bounded by the curve, the x-axis, and the line x = 3.
Consider the function y = \left(x + 1\right)^{3} + 2.
Sketch the graph of the function.
Hence, determine the exact area bounded by the curve, the x-axis, and the lines x = - 2 and x = 1.
Consider the function y = \left(x - 1\right)^{2} \left(x + 3\right).
Sketch the graph of the function.
Hence, determine the exact area bounded by the curve, the x-axis and the lines \\ x = - 1 and x = 2.
Consider the function y = 2 x + 3.
Sketch the graph of the function.
Calculate the exact area bounded by the curve, x = 1, x = 3 and the x-axis.
Consider the function y = - 2 x + 8.
Sketch the graph of the function.
Calculate the exact area bounded by the curve, x = 1, x = 3 and the x-axis.
Calculate \int_{3}^{9} \left(x - 3\right) \left(x - 9\right)\, dx.
Hence, determine the area bounded by the curve y = \left(x - 3\right) \left(x - 9\right), the x-axis and the bounds x = 3 and x = 9.
Calculate \int_{ - 4 }^{3} 5\, dx.
Hence, determine the area bounded by the curve y = 5, the x-axis and the lines x = - 4 and x = 3.
Calculate \int_{ - 2 }^{4} \left( 2 x - 8\right)\, dx.
Hence, determine the area bounded by the line y = 2 x - 8, the x-axis and the lines \\ x = - 2 and x = 4.
Calculate \int_{ - 8 }^{ - 2 } - \left(x + 2\right) \left(x + 8\right) \, dx.
Hence, determine the area bounded by the curve y = - \left(x + 2\right) \left(x + 8\right), the x-axis and the lines x = - 8 and x = - 2.
Evaluate the following:
Consider the function f \left( t \right) = t^{4} + 8 t^{2} + 20. Find \int_{1}^{x} f \left( t \right)\, dt.
The function f has an antiderivative F, and F \left( 3 \right) = 4.
Express \int_{3}^{x} f \left( t \right)\, dt in terms of F and x.
Find \dfrac{d}{d x} \int_{3}^{x} f \left( t \right) dt.
Consider the function f \left( t \right) = - 4 t.
Find \int_{6}^{x} f \left( t \right)\, dt.
Hence, find \dfrac{d}{d x} \int_{6}^{x} f \left( t \right)\, dt.
What can we conclude about \dfrac{d}{d x} \int_{6}^{x} f \left( t \right)\, dt?
Consider the function f \left( t \right) = 12 t + 9.
Find \int_{ - 3 }^{x} f \left( t \right)\, dt.
Hence, find \dfrac{d}{d x} \int_{ - 3 }^{x} f \left( t \right)\, dt.
Consider the expression \dfrac{d}{d x} \int_{k}^{x} \dfrac{1}{\sqrt{t}}\, dt.
What restrictions must be on the value of k for the integration to be possible?
Find \int_{k}^{x} \dfrac{1}{\sqrt{t}}\, dt.
Hence, find \dfrac{d}{d x} \int_{k}^{x} \dfrac{1}{\sqrt{t}}\, dt.