By setting up a formula for Riemann sums, andthen taking their limit, calculate1|(2x-x³).-1Use only the definition of the definite integralgiven in the Reading Supplement for Section5.3.Do not use the Fundamental Theorem ofCalculus, or attempt a geometric argument.Show ALL of your work and Explanation Step-by-Step In this section we consider the limit of general Riemann sums as the norm of the partitionsof a closed interval [a, b] approaches zero. This limiting process leads us to the definitionof the definite integral of a function over a closed interval [a, b].5.3 The Definite Integral319Definition of the Definite IntegralWhen the definite integral exișts, we say that the Riemann sums of ƒ on [a, b] convergeto the definite integral J = ["f(x) dx and that f is integrable over [a, b].In the cases where the subintervals all have equal width Ax = (b – a)/n, the Rie-The definition of the definite integral is based on the fact that for some functions, as thenorm of the partitions of [ a, b] approaches zero, the values of the corresponding Riemannmann sums have the form= E f(c) Ax, =Ax = Ar = (b – a)/n for all kk-1k-1where c is chosen in the kth subinterval. If the definite integral exists, then these Riemannsums converge to the definite integral of f over [a, b], soь — аsums approach a limiting value J. We introduce the symbol ɛ as a small positive numberthat specifies how close to J the Riemann sum must be, and the symbol 8 as a secondsmall positive number that specifies how small the norm of a partition must be in order forconvergence to happen. We now define this limit precisely.For equal-width subintervals,||P| 0 is the same as n- 0.J =f(x)dx = limk-1If we pick the point c to be the right endpoint of the kth subinterval, so thatC = a + kAx =a + k(b – a)/n, then the formula for the definite integral becomesDEFINITION Let f(x) be a function defined on a closed interval [a, b]. Wesay that a number J is the definite integral of f over [a, b] and that J is the limitof the Riemann sums E- f(c) Ax, if the following condition is satisfied:Given any number e 0 there is a corresponding number 8 0 suchthat for every partition P = {xg, X1. . .. ,x,} of [ a, b] with || P ||

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By setting up a formula for Riemann sums, andthen taking their limit, calculate1|(2x-x³).-1Use only the definition of the definite integralgiven in the Reading Supplement for Section5.3.Do not use the Fundamental Theorem ofCalculus, or attempt a geometric argument.Show ALL of your work and Explanation Step-by-Step In this section we consider the limit of general Riemann sums as the norm of the partitionsof a closed interval [a, b] approaches zero. This limiting process leads us to the definitionof the definite integral of a function over a closed interval [a, b].5.3 The Definite Integral319Definition of the Definite IntegralWhen the definite integral exișts, we say that the Riemann sums of ƒ on [a, b] convergeto the definite integral J = [&#34;f(x) dx and that f is integrable over [a, b].In the cases where the subintervals all have equal width Ax = (b – a)/n, the Rie-The definition of the definite integral is based on the fact that for some functions, as thenorm of the partitions of [ a, b] approaches zero, the values of the corresponding Riemannmann sums have the form= E f(c) Ax, =Ax = Ar = (b – a)/n for all kk-1k-1where c is chosen in the kth subinterval. If the definite integral exists, then these Riemannsums converge to the definite integral of f over [a, b], soь — аsums approach a limiting value J. We introduce the symbol ɛ as a small positive numberthat specifies how close to J the Riemann sum must be, and the symbol 8 as a secondsmall positive number that specifies how small the norm of a partition must be in order forconvergence to happen. We now define this limit precisely.For equal-width subintervals,||P| 0 is the same as n- 0.J =f(x)dx = limk-1If we pick the point c to be the right endpoint of the kth subinterval, so thatC = a + kAx =a + k(b – a)/n, then the formula for the definite integral becomesDEFINITION Let f(x) be a function defined on a closed interval [a, b]. Wesay that a number J is the definite integral of f over [a, b] and that J is the limitof the Riemann sums E- f(c) Ax, if the following condition is satisfied:Given any number e > 0 there is a corresponding number 8 > 0 suchthat for every partition P = {xg, X1. . .. ,x,} of [ a, b] with || P || < 8 and anychoice of c in [X-1. X], we haveA Formula for the Riemann Sum with Equal-Width Subintervalsf(x) dx = limb+ k(1)flo) Ax, --1Equation (1) gives one explicit formula that can be used to compute definite integrals. Aslong as the definite integral exists, the Riemann sums corresponding to other choices ofpartitions and locations of points c̟ will have the same limit as n–∞, provided that thenorm of the partition approaches zero.The value of the definite integral of a function over any particular interval depends onthe function, not on the letter we choose to represent its independent variable. If we decideto use t or u instead of x, we simply write the integral asThe definition involves a limiting process in which the norm of the partition goes to zero.We have many choices for a partition P with norm going to zero, and many choices ofpoints c for each partition. The definite integral exists when we always get the same limitJ, no matter what choices are made. When the limit exists we writelimEfla) Ax,f(1) dtf(u) duinstead oforI we say that the definite integral exists. The limit of any Riemann sum is always takenas the norm of the partitions approaches zero and the number of subintervals goes to infin-ity, and furthermore the same limit J must be obtained no matter what choicesandNo matter how we write the integral, it is still the same number, the limit of the Riemannsums as the norm of the partition approaches zero. Since it does not matter what letter weuse, the variable of integration is called a dummy variable. In the three integrals givenabove, the dummy variables are 1, u, and x.make forthe points c-Leibniz introduced a notation for the definite integral that captures its construction asa limit of Riemann sums. He envisioned the finite sums E- f(c) Ax, becoming an infi-nite sum of function values f(x) multiplied by &#34;infinitesimal&#34; subinterval widths dx. Thesum symbol E is replaced in the limit by the integral symbol /, whose origin is in theletter &#34;S&#34; (for sum). The function values f(c) are replaced by a continuous selection offunction values f(x). The subinterval widths Ax, become the differential dx. It is as if weare summing all products of the form f(x)· dx as x goes from a to b. While this notationcaptures the process of constructing an integral, it is Riemann's definition that gives a pre-cise meaning to the definite integral.If the definite integral exists, then instead of writing J we writef(x) dx.We read this as &#34;the integral from a to b of f of x dee x&#34; or sometimes as &#34;the integral froma to b of f of x with respect to x.&#34; The component parts in the integral symbol also havenames:The function f(x) is the integrand.Upper limit of integrationb'x is the variable of integration.Integral signf(x) dxJaWhen you find the valueof the integral, you haveevaluated the integral.Lower limit of integrationIntegral of f from a to b

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By setting up a formula for Riemann sums, ar then taking their limit, calculate 1 Jax |(2x-x'). -1