Divide the interval [a,b] into n equal subintervals
[x_{0}, x_{1}], [x_{1}, x_{2}].................,[x_{n-2}, x_{n-1}], [x_{n-1}, x_{n}]
such that a = x_{0} < x_{1} < x_{2 }< .............< x_{n-1}< x_{n} < b.
Evaluate the following integrals as the limits of sums:
Problem 1 :
Integral 0 to 1 (5x+4) dx
Solution :
f(x) = (5x+4), a = 0 and b = 1
f(a+(b-a)(r/n)) = f(0+(1-0)(r/n))
= f(r/n)
By applying the limit
= (5/2) lim_{ n->}_{∞} (1+1/n)
= (5/2)(1+0)
= 5/2
Problem 2 :
Integral 1 to 2 (4x^{2}-1) dx
Solution :
f(x) = (5x+4), a = 1 and b = 2
f(a+(b-a)(r/n)) = f(1+(2-1)(r/n))
= f(1+r/n)
f(1+(r/n)) = (4(1+(r/n))^{2}-1)
= 4(1+r^{2}/n+2r/n) - 1
= 3+4r^{2}/n+8r/n
By applying the limits,
= 3+(2/3)(2) + 4
= 3+4/3 + 4
= 7+(4/3)
= (21+4)/3
= 25/3
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