Mathematics Foundations/10.1 Definite and Indefinite Integrals

10.1 Definite and Indefinite Integrals 10.1.1 Introduction to Integration Integration is one of the fundamental operations in calculus, complementary to differentiation. While differentiation measures the rate of change of a function, integration measures the accumulation of quantities. 10.1.2 Indefinite Integrals 10.1.2.1 Definition The indefinite integral of a function , denoted by , represents the family of all antiderivatives of . An antiderivative of is a function such that . where is an arbitrary constant called the constant of integration. 10.1.2.2 Basic Integration Rules 1. , where is a constant 2. , for 3. 4. 5. 6. 7. 8. 9. 10. 10.1.2.3 Linearity of Integration Integration is a linear operation, which means: 1. 2. , where is a constant 10.1.3 Definite Integrals 10.1.3.1 Definition The definite integral of a function from to , denoted by , represents the signed area between the curve and the x-axis from to . 10.1.3.2 Riemann Sums The definite integral can be defined as the limit of Riemann sums: where the interval is divided into subintervals, is the width of the -th subinterval, and is a point in the -th subinterval. For regular partitions with equal subinterval widths: (for midpoint rule) 10.1.3.3 Properties of Definite Integrals 1. 2. 3. 4. , where is a constant 5. , for any between and 6. If for all , then 7. If for all , then 8. 10.1.4 The Fundamental Theorem of Calculus The Fundamental Theorem of Calculus establishes the relationship between differentiation and integration. 10.1.4.1 First Part If is continuous on and , then for all . 10.1.4.2 Second Part If is continuous on and is any antiderivative of on , then: This is often written using the notation: 10.1.5 Applications of Integration 10.1.5.1 Area Between Curves The area between two curves and from to , where for all , is given by: 10.1.5.2 Volume of Solids of Revolution The volume of a solid formed by revolving the region bounded by , the x-axis, and the lines and around the x-axis is: For revolution around the y-axis: 10.1.5.3 Arc Length The length of a curve from to is: 10.1.5.4 Surface Area The surface area of the solid formed by revolving the curve from to around the x-axis is: 10.1.6 Mean Value of a Function The average or mean value of a function over the interval is: 10.1.7 Exercises 1. Evaluate the indefinite integral . 2. Find using the Fundamental Theorem of Calculus. 3. Calculate the area between the curves and from to . 4. Find the volume of the solid obtained by rotating the region bounded by , , and about the x-axis. 5. Determine the average value of over the interval . 6. Evaluate . 7. Find the area of the region bounded by , , , and .