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- Maths and statistics
- Integration
Antidifferentiation is the reverse operation of differentiation. You need it to solve problems involving area under a curve, total accumulated change, and reversing rates of change. By mastering antidifferentiation, you'll be able to reconstruct original functions and solve real-world problems in physics, engineering, and beyond. If you have the derivative...
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- Maths and statistics
- Integration
Integrals can be used to find the area between a curve and the \(x\)-axis. We can use this to calculate the total accumulation of a quantity in physics, economics and engineering. Use this resource to learn how to calculate area under the curve. Finding the area under a curve Consider...
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- Maths and statistics
- Integration
What if you have an area that is bounded by limits on both the \(x\) and the \(y\)-axes? This is where double integrals come into play. We use double integrals in physics for finding mass in a region given a density function, engineering for determining properties of materials with varying...
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Integration is a fundamental concept in calculus that involves finding the total accumulated change or area under a curve. It is used to calculate distances from velocity, determine areas and volumes, and solve differential equations. Integration helps us understand the cumulative effects of changing quantities, from determining total growth to...
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- Maths and statistics
- Integration
Integration by parts is used to solve integrals by expressing them in terms of simpler integrals. This is applied in areas like mechanical engineering for calculating work done by variable forces, in physics for solving complex integrals, and in economics for evaluating growth models. Use this resource to learn how...
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- Maths and statistics
- Integration
Integration by substitution simplifies the process of finding integrals by making a change of variables. This is used for calculating areas and volumes in engineering, solving differential equations in physics, and optimising functions in economics. Use this resource to learn how to integrate by substitution. An expression composed of two...
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- Maths and statistics
- Integration
How do you integrate an exponential function? This skill is important for calculating compound interest over time in finance, determining population growth models in biology, and analysing radioactive decay in physics. Use this resource to learn how. Exponential functions have the form: \[y=ae^{kx}\] where \(a\) and \(k\) are constants. Examples...
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- Maths and statistics
- Integration
How do you integrate polynomial functions, where different terms have different powers and coefficients? This skill is important for calculating projectile trajectories in physics, beam deflections in engineering and determining cost functions and profit maximisation in economics. Use this resource to learn how. In Antidifferentiation, you learned how to find...
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- Maths and statistics
- Integration
How do you integrate a reciprocal function? This skill is important for calculating electrical currents in circuits and modelling harmonic motion in mechanical systems. Use this resource to learn how. Reciprocal functions have the form: \[y=\frac{k}{x}\] where \(k\) is a constant. We often deal with reciprocal functions when looking at...
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- Maths and statistics
- Integration
How do you integrate a trigonometic function? This skill is important for solving problems involving wave patterns in physics, analysing alternating current circuits in engineering and determining structural loads with curved geomoetries in architecture. Use this resource to learn how. You already learnt about the general forms of Circular functions....
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- Maths and statistics
- Integration
How do you integrate an expression when there is an algebraic expression in the numerator and denominator of a fraction? Integrating using partial fractions helps you to solve this problem. Use this resource to learn how to integrate using partial fractions. Sometimes a complex function may be integrated by breaking...
