Understanding the Fundamental Theorem of Calculus
Explore the connection between differential calculus and the definite integral through the fundamental theorem of calculus, which allows for the evaluation of complex summations. Discover the properties of definite integrals and how to apply the theorem to find areas under curves. Practice evaluating definite integrals analytically and using a graphical display calculator with examples.
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24 September 2024 The fundamental Theorem of Calculus LO: To evaluate definite integrals analytically and using the GDC. www.mathssupport.org
Fundamental theorem of calculus Isaac Newton and Gottfried Leibniz independently came to realize that differential calculus and the definite integral are linked together. This link is called the Fundamental theorem of calculus. The beauty of this theorem is that it enables us to evaluate complicated summations. It is based on the fact that the quotient ? a secant line, gives us an approximation for the slope of a tangent line. ?, the slope of The product ( y)( x), the area of a rectangle, gives us an approximation for the area under a curve. www.mathssupport.org
Fundamental theorem of calculus This fact is established in the following theorem. Fundamental theorem of calculus If f is a continuous function on the interval a x b and F is an antiderivative of f on a x b, then ? = ?(?)? = F(b) F(a) ? ? ? ? ? The notation ?(?)? ?means F(b) F(a) www.mathssupport.org
Properties of the definite integral The following properties of the definite integral can all be deduced from the Fundamental theorem of calculus. ? ? ? ? = 0 ? ? ? = ? ? ? ? ? ? ? ? ? ? ?? ? ? = ? ? ? ? ? ? ? ? ? ?(?) ?(?) ? = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? + ? ? ? ? ? ? ? = ?(b a) ? ? www.mathssupport.org ?
Fundamental theorem of calculus Use the fundamental theorem of calculus to find the area: between the x-axis and y = x2 from x = 0 to x = 1 f(x) = x2 y=?2 F(x) = ?3 So the area has antiderivative Is the antiderivative 3 1 ?2 ? = 0 x 0 1 =F(1) F(0) =13 3 03 =1 3 3 www.mathssupport.org
Fundamental theorem of calculus Use the fundamental theorem of calculus to find the area: between the x-axis and y = ? from x = 1 to x = 9 f(x) = ? y= ? F(x) = ? So the area 1 2 has antiderivative = ? 3 2 =2 Is the antiderivative 3? ? 3 2 9 1 2 ? = ? 1 x 9 1 =F(9) F(1) =2 39 9 2 =2 31 1 3 (1) 3 (27) 2 = 171 3 www.mathssupport.org
Fundamental theorem of calculus 1 Evaluate the definite integral ? 1 ? 2 1 1 2?2 ? Find the simplest antiderivative of x - 1 2 Evaluate the antiderivative at x = 1 and x = -2, then find the difference 1 2(12) 1 1 2( 2)2 ( 2) = 1 2 1 2 + 2 = = 1 2 4 = 9 2 www.mathssupport.org
Fundamental theorem of calculus 1 Evaluate the definite integral Recall that ?? + ?? ? = 1 ? 2? 33 ? 1 1 1 2 1 42? 34 1 ? + 1?? + ??+1 + ? 1 Evaluate the antiderivative at x = 1 and x = -1, then find the difference 1 8(2(1) ) 1 8(2( 1) ) - 34 34 = 1 8 625 = 8 = 78 www.mathssupport.org
Fundamental theorem of calculus 5 ?2?+1 Evaluate the definite integral ?2 ? 1 5 ?2?+ ? 2 ? 1 5 Recall that ???+? ? = 1 ????+?+ ? 1 2?2? 1 ?1 Evaluate the antiderivative at x = 5 and x = 1, then find the difference 1 2(?2(5) 1 1 2(?2(1) 1 = 5 1 =1 2?10 1 2?2+4 5 =5?10 5?2+8 10 www.mathssupport.org
Evaluating definite integrals using GDC www.mathssupport.org
Fundamental theorem of calculus 41 ? Use a GDC to evaluate the definite integral ? ? 1 Turn on the GDC Press the math button www.mathssupport.org
Fundamental theorem of calculus 41 ? Use a GDC to evaluate the definite integral ? ? 1 Turn on the GDC Press the math button Press 9 www.mathssupport.org
Fundamental theorem of calculus 41 ? Use a GDC to evaluate the definite integral ? ? 1 Turn on the GDC Press the math button Press 9 Press www.mathssupport.org
Fundamental theorem of calculus 41 ? Use a GDC to evaluate the definite integral ? ? 1 Turn on the GDC Press the math button Press 9 Press 41 ? = 1 ? ? 1 www.mathssupport.org
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