2 interactive labs · AS & A Level Mathematics 9709
Antiderivatives: The Family y = F(x) + c
1.8 Integration · Integrate (ax + b)ⁿ and check by differentiating, slide c to see a whole family of curves, then fix c with a given point.
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About this lab
Integrate (ax + b)ⁿ and check by differentiating, slide c to see a whole family of curves, then fix c with a given point.
You will be able to
Integrate m(ax + b)ⁿ for any rational n except −1, and sums and differences of such terms, dividing by the new power and by a.
Check an integral by differentiating it, and explain why + c is needed: every curve of the family has the same gradient at the same x.
Find the constant of integration, exactly, so that a curve with a given dy/dx passes through a given point, and state its equation.
Signed Integral or Geometric Area?
1.8 Integration · Drag the limits across the x-axis, or between two graphs, and predict: when does ∫ differ from the area, and when is an improper integral finite?
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About this lab
Drag the limits across the x-axis, or between two graphs, and predict: when does ∫ differ from the area, and when is an improper integral finite?
You will be able to
Evaluate a definite integral exactly and explain why the part of the region below the x-axis counts as negative.
Find the area of a region that crosses the x-axis by splitting at the roots and adding the absolute values, and the area between a curve and a line or between two curves as ∫(upper − lower).
Evaluate a simple improper integral such as ∫₀¹ x^(−1/2) dx as a limit and state its value.
