<prompt> Hlalosa lipalo tse tšehetsang mohopolo oa integral calculus.nan  <response>: Instruction in English: Describe the mathematics behind the concept of integral calculus. . Response in English: Integral calculus, also known as integration, is a branch of calculus that deals with the accumulation of quantities and the calculation of areas, volumes and the net change. It is the inverse operation to differentiation, or finding the derivative of a function.The fundamental theorem of calculus establishes the mathematical link between differentiation and integration. It states that differentiation and integration are inverse operations, and that the definite integral of a function can be found by evaluating its antiderivative, also known as the indefinite integral, at the endpoints of the interval over which the integration is taking place.The concept of integration can be visualized through the use of Riemann sums. In this method, the area under the curve of a function is approximated by dividing the region into a number of equally spaced vertical intervals, and calculating the sum of the areas of rectangles with heights determined by the function value at a specific point within each interval. As the number of intervals increases, the approximation becomes more accurate, and when the number of intervals approaches infinity, the Riemann sum becomes equal to the definite integral of the function.Integration can be used to calculate a variety of physical quantities, including the area under a curve, the volume of a solid of revolution, the work done by a variable force, and the average value of a function. It can also be used to solve differential equations, which involve calculating a function based on its derivative.Overall, integral calculus is a powerful mathematical tool that allows for the quantitative analysis of continuous change and the accumulation of quantities. . Response in Sesotho: Integral calculus, e tsejoang hape e le kopanyo, ke lekala la lipalo le sebetsanang le ho bokella lipalo le ho bala libaka, li-volumes le ho fetoha ha nete. Ke tsebetso e fapaneng ho ea karohano, kapa ho fumana motsu oa tšebetso. Theorem ea motheo ea calculus e theha kamano ea lipalo pakeng tsa phapang le kopanyo. E bolela hore phapang le kopanyo ke tsebetso e fapaneng, le hore karolo e hlakileng ea tsebetso e ka fumanoa ka ho hlahloba antiderivative ea eona, e tsejoang hape e le motsoako o sa lekanyetsoang, qetellong ea nako eo ho kopanngoa ho etsoang ho eona. Khopolo ea ho kopanya e ka bonoa ka tšebeliso ea lipalo tsa Riemann. Ka mokhoa ona, sebaka se ka tlasa mokokotlo oa tsebetso se hakanyetsoa ka ho arola sebaka seo ka makhetlo a mangata a lekanang, me ho baloa kakaretso ea libaka tsa khutlonnetsepa tse nang le bophahamo bo lekantsoeng ke boleng ba tsebetso sebakeng se itseng ka nako e itseng. . Ha palo ya dikgao e ntse e eketseha, kakanyo e ba e nepahetseng, mme ha palo ya dikgao e atamela ho sa feleng, kakaretso ya Riemann e ba e lekanang le kakaretso e hlakileng ya mosebetsi. Ho kopanya ho ka sebelisoa ho bala mefuta e sa tšoaneng ea mele, ho kenyelletsa le sebaka se ka tlasa lekhalo, bophahamo ba molumo o tiileng oa phetoho, mosebetsi o entsoeng ka matla a feto-fetohang, le boleng bo tloaelehileng ba mosebetsi. E ka boela ea sebelisoa ho rarolla li-equations tse fapaneng, tse kenyelletsang ho bala mosebetsi ho latela motsoako oa eona. Ka kakaretso, integral calculus ke sesebelisoa se matla sa lipalo se lumellang tlhahlobo ea palo ea liphetoho tse tsoelang pele le ho bokella lipalo. 